Abrasive article including shaped abrasive particles

ABSTRACT

A shaped abrasive particle including a body comprising a first major surface, a second major surface, and a side surface extending between the first major surface and the second major surface, the body comprising a sharpness-shape-strength factor (3SF) within a range between about 0.7 and about 1.7 and a Shape Index within a range between at least about 0.01 and not greater than about 0.49.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application is a continuation of and claims priority to U.S. patentapplication Ser. No. 16/745,661, entitled “ABRASIVE ARTICLE INCLUDINGSHAPED ABRASIVE PARTICLES,” by Christopher ARCONA et al., filed Jan. 17,2020, which is a continuation of and claims priority to U.S. patentapplication Ser. No. 14/686,733, entitled “ABRASIVE ARTICLE INCLUDINGSHAPED ABRASIVE PARTICLES,” by Christopher ARCONA et al., filed Apr. 14,2015, now U.S. Pat. No. 10,557,067, which claims priority under 35U.S.C. § 119(e) to U.S. Provisional Application No. 61/979,457, entitled“ABRASIVE ARTICLE INCLUDING SHAPED ABRASIVE PARTICLES,” by ChristopherARCONA et al., filed Apr. 14, 2014, which are all assigned to thecurrent assignee hereof and incorporated herein by reference in theirentireties.

BACKGROUND Field of the Disclosure

The following is directed to abrasive articles, and particularly,abrasive articles including shaped abrasive particles.

Description of the Related Art

Abrasive particles and abrasive articles made from abrasive particlesare useful for various material removal operations including grinding,finishing, and polishing. Depending upon the type of abrasive material,such abrasive particles can be useful in shaping or grinding a widevariety of materials and surfaces in the manufacturing of goods. Certaintypes of abrasive particles have been formulated to date that haveparticular geometries, such as triangular shaped abrasive particles andabrasive articles incorporating such objects. See, for example, U.S.Pat. Nos. 5,201,916; 5,366,523; and 5,984,988.

Three basic technologies that have been employed to produce abrasiveparticles having a specified shape are (1) fusion, (2) sintering, and(3) chemical ceramic. In the fusion process, abrasive particles can beshaped by a chill roll, the face of which may or may not be engraved, amold into which molten material is poured, or a heat sink materialimmersed in an aluminum oxide melt. See, for example, U.S. Pat. No.3,377,660 (disclosing a process including flowing molten abrasivematerial from a furnace onto a cool rotating casting cylinder, rapidlysolidifying the material to form a thin semisolid curved sheet,densifying the semisolid material with a pressure roll, and thenpartially fracturing the strip of semisolid material by reversing itscurvature by pulling it away from the cylinder with a rapidly drivencooled conveyor).

In the sintering process, abrasive particles can be formed fromrefractory powders having a particle size of up to 10 micrometers indiameter. Binders can be added to the powders along with a lubricant anda suitable solvent, e.g., water. The resulting mixture, mixtures, orslurries can be shaped into platelets or rods of various lengths anddiameters. See, for example, U.S. Pat. No. 3,079,242 (disclosing amethod of making abrasive particles from calcined bauxite materialincluding (1) reducing the material to a fine powder, (2) compactingunder affirmative pressure and forming the fine particles of said powderinto grain sized agglomerations, and (3) sintering the agglomerations ofparticles at a temperature below the fusion temperature of the bauxiteto induce limited recrystallization of the particles, whereby abrasivegrains are produced directly to size).

Chemical ceramic technology involves converting a colloidal dispersionor hydrosol (sometimes called a sol), optionally in a mixture, withsolutions of other metal oxide precursors, into a gel or any otherphysical state that restrains the mobility of the components, drying,and firing to obtain a ceramic material. See, for example, U.S. Pat.Nos. 4,744,802 and 4,848,041. Other relevant disclosures on shapedabrasive particles and associated methods of forming and abrasivearticles incorporating such particles are available at:abelip.com/publications/.

Still, there remains a need in the industry for improving performance,life, and efficacy of abrasive particles, and the abrasive articles thatemploy abrasive particles.

SUMMARY

According to one aspect, a method of making a shaped abrasive particleincludes forming a body of a shaped abrasive comprising at least one ofi) selecting a material having a predetermined strength within a rangebetween at least about 350 MPa and not greater than about 1500 MPa andforming the body of the shaped abrasive particle with a predeterminedtip sharpness and predetermined Shape Index based upon the predeterminedstrength, ii) selecting a predetermined Shape Index of the body of theshaped abrasive particle within a range between at least about 0.01 andnot greater than about 0.49 and forming the body with the predeterminedtip sharpness and the predetermined strength based upon thepredetermined Shape Index, and iii) selecting a predetermined tipsharpness of a body of the shaped abrasive particle within a rangebetween at least about 1 micron and not greater than about 80 microns,and forming the body of a shaped abrasive particle with a predeterminedShape Index and a predetermined strength based upon the predeterminedtip sharpness.

In one aspect, a shaped abrasive particle includes a body comprising afirst major surface, a second major surface, and a side surfaceextending between the first major surface and the second major surface,wherein the body comprises a sharpness-shape-strength factor (3SF)within a range between about 0.7 and about 1.7 and a Shape Index withina range between at least about 0.01 and not greater than about 0.49.

For another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises a Shape Index within a range betweenat least about 0.01 and not greater than about 0.49 and a strengthwithin a range between at least about 350 MPa and not greater than about1500 MPa.

According to another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises an average tip sharpness within arange between not greater than about 80 microns and at least about 1micron, a Shape Index within a range between at least about 0.01 and notgreater than about 0.49, and the body comprises a strength of at leastabout 350 MPa and not greater than about 1500 MPa.

For another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein a first portion of the side surface has apartially-concave shape.

In still another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein a first portion of the side surface extends between afirst corner and second corner of the body that are adjacent to eachother, and wherein the first portion of the side surface comprises afirst curved section joined to a first linear section.

According to yet another aspect, a shaped abrasive particle includes abody comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein a first portion of the side surface comprises afirst curved section joined to a first linear section and defining aninterior corner defining an obtuse angle.

For another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body is a hybrid polygonal shape having a sum ofthe external corners of substantially 180 degrees, and furthercomprising a first portion of the side surface having a first curvedsection.

In another aspect, a shaped abrasive particle includes a body comprisinga first major surface, a second major surface, and a side surfaceextending between the first major surface and the second major surface,wherein the body comprises a first maximum tip width (Wt1) that definesa maximum width of a first arm of the body, and wherein the firstmaximum tip width (Wt1) is disposed a distance from a first terminal endof the first arm and between a midpoint of the body and the firstterminal end.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure may be better understood, and its numerousfeatures and advantages made apparent to those skilled in the art byreferencing the accompanying drawings.

FIG. 1 includes a portion of a system for forming a particulate materialin accordance with an embodiment.

FIG. 2 includes a portion of the system of FIG. 1 for forming aparticulate material in accordance with an embodiment.

FIG. 3 includes a cross-sectional illustration of a shaped abrasiveparticle for illustration of certain features according to embodiments.

FIG. 4 includes a side view of a shaped abrasive particle and percentageflashing according to an embodiment.

FIG. 5A includes an illustration of a bonded abrasive articleincorporating shaped abrasive particles in accordance with anembodiment.

FIG. 5B includes a cross-sectional illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 6 includes a cross-sectional illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 7 includes a top-down illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 8A includes a top-down illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 8B includes a perspective view illustration of a portion of acoated abrasive article according to an embodiment.

FIG. 9 includes a perspective view illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 10 includes a top view illustration of a portion of an abrasivearticle in accordance with an embodiment.

FIG. 11 includes images representative of portions of a coated abrasiveaccording to an embodiment and used to analyze the orientation of shapedabrasive particles on the backing.

FIG. 12A includes a perspective view illustration of a shaped abrasiveparticle according to an embodiment.

FIG. 12B includes a top view illustration of a shaped abrasive particleaccording to an embodiment.

FIG. 13 includes a top view illustration of a shaped abrasive particleaccording to an embodiment.

FIG. 14 includes a top view illustration of a shaped abrasive particleaccording to an embodiment.

FIG. 15 includes a top view illustration of a shaped abrasive particleaccording to an embodiment.

FIG. 16 includes a top view illustration of a shaped abrasive particleaccording to an embodiment.

FIG. 17 includes a top view illustration of a shaped abrasive particleaccording to alternative embodiment.

FIG. 18A includes a top view of a shaped abrasive particle according toan embodiment.

FIG. 18B includes a top view of a shaped abrasive particle according toan embodiment.

FIG. 18C includes a top-down image of a shaped abrasive particle with aline of sectioning for measurement of draft angle according to anembodiment.

FIG. 18D includes a cross-sectional image of a shaped abrasive particlefor measurement of a draft angle according to an embodiment.

FIG. 18E includes a cross-sectional image of a shaped abrasive particlefor measurement of a draft angle according to an embodiment.

FIG. 19 includes a generalized plot of specific grinding energy versuscumulative material removed for three coated abrasive articles.

FIG. 20 includes an image of a shaped abrasive particle according to anExample.

FIG. 21 includes an image of a shaped abrasive particle according to anExample.

FIG. 22 includes an image of a shaped abrasive particle according to anExample.

FIG. 23 includes an image of a conventional shaped abrasive particle.

FIG. 24 includes an image of a shaped abrasive particle according to anExample.

FIG. 25 includes an image of a shaped abrasive particle.

FIG. 26 includes a plot of median force per total area removed from theworkpiece for shaped abrasive particles of the Examples.

FIG. 27 includes a plot of specific grinding energy per cumulativematerial removed for a conventional sample and a representative sample

DETAILED DESCRIPTION

The following is directed to abrasive articles including shaped abrasiveparticles. The methods herein may be utilized in forming shaped abrasiveparticles and using abrasive articles incorporating shaped abrasiveparticles. The shaped abrasive particles may be utilized in variousapplications, including for example coated abrasives, bonded abrasives,free abrasives, and a combination thereof. Various other uses may bederived for the shaped abrasive particles.

Shaped Abrasive Particles

Various methods may be utilized to obtain shaped abrasive particles. Theparticles may be obtained from a commercial source or fabricated. Somesuitable processes used to fabricate the shaped abrasive particles caninclude, but is not limited to, depositing, printing (e.g.,screen-printing), molding, pressing, casting, sectioning, cutting,dicing, punching, pressing, drying, curing, coating, extruding, rolling,and a combination thereof.

FIG. 1 includes an illustration of a system 150 for forming a shapedabrasive particle in accordance with one, non-limiting embodiment. Theprocess of forming shaped abrasive particles can be initiated by forminga mixture 101 including a ceramic material and a liquid. In particular,the mixture 101 can be a gel formed of a ceramic powder material and aliquid. In accordance with an embodiment, the gel can be formed of theceramic powder material as an integrated network of discrete particles.

The mixture 101 may contain a certain content of solid material, liquidmaterial, and additives such that it has suitable rheologicalcharacteristics for use with the process detailed herein. That is, incertain instances, the mixture can have a certain viscosity, and moreparticularly, suitable rheological characteristics that form adimensionally stable phase of material that can be formed through theprocess as noted herein. A dimensionally stable phase of material is amaterial that can be formed to have a particular shape and substantiallymaintain the shape for at least a portion of the processing subsequentto forming. In certain instances, the shape may be retained throughoutsubsequent processing, such that the shape initially provided in theforming process is present in the finally-formed object. It will beappreciated that in some instances, the mixture 101 may not be ashape-stable material, and the process may rely upon solidification andstabilization of the mixture 101 by further processing, such as drying.

The mixture 101 can be formed to have a particular content of solidmaterial, such as the ceramic powder material. For example, in oneembodiment, the mixture 101 can have a solids content of at least about25 wt %, such as at least about 35 wt %, or even at least about 38 wt %for the total weight of the mixture 101. Still, in at least onenon-limiting embodiment, the solids content of the mixture 101 can benot greater than about 75 wt %, such as not greater than about 70 wt %,not greater than about 65 wt %, not greater than about 55 wt %, notgreater than about 45 wt %, or not greater than about 42 wt %. It willbe appreciated that the content of the solids materials in the mixture101 can be within a range between any of the minimum and maximumpercentages noted above.

According to one embodiment, the ceramic powder material can include anoxide, a nitride, a carbide, a boride, an oxycarbide, an oxynitride, anda combination thereof. In particular instances, the ceramic material caninclude alumina. More specifically, the ceramic material may include aboehmite material, which may be a precursor of alpha alumina. The term“boehmite” is generally used herein to denote alumina hydrates includingmineral boehmite, typically being Al₂O₃.H₂O and having a water contenton the order of 15%, as well as pseudoboehmite, having a water contenthigher than 15%, such as 20-38% by weight. It is noted that boehmite(including pseudoboehmite) has a particular and identifiable crystalstructure, and therefore a unique X-ray diffraction pattern. As such,boehmite is distinguished from other aluminous materials including otherhydrated aluminas such as ATH (aluminum trihydroxide), a commonprecursor material used herein for the fabrication of boehmiteparticulate materials.

Furthermore, the mixture 101 can be formed to have a particular contentof liquid material. Some suitable liquids may include water. Inaccordance with one embodiment, the mixture 101 can be formed to have aliquid content less than the solids content of the mixture 101. In moreparticular instances, the mixture 101 can have a liquid content of atleast about 25 wt % for the total weight of the mixture 101. In otherinstances, the amount of liquid within the mixture 101 can be greater,such as at least about 35 wt %, at least about 45 wt %, at least about50 wt %, or even at least about 58 wt %. Still, in at least onenon-limiting embodiment, the liquid content of the mixture can be notgreater than about 75 wt %, such as not greater than about 70 wt %, notgreater than about 65 wt %, not greater than about 62 wt %, or even notgreater than about 60 wt %. It will be appreciated that the content ofthe liquid in the mixture 101 can be within a range between any of theminimum and maximum percentages noted above.

Furthermore, to facilitate processing and forming shaped abrasiveparticles according to embodiments herein, the mixture 101 can have aparticular storage modulus. For example, the mixture 101 can have astorage modulus of at least about 1×10⁴ Pa, such as at least about 4×10⁴Pa, or even at least about 5×10⁴ Pa. However, in at least onenon-limiting embodiment, the mixture 101 may have a storage modulus ofnot greater than about 1×10⁷ Pa, such as not greater than about 2×10⁶Pa. It will be appreciated that the storage modulus of the mixture 101can be within a range between any of the minimum and maximum valuesnoted above.

The storage modulus can be measured via a parallel plate system usingARES or AR-G2 rotational rheometers, with Peltier plate temperaturecontrol systems. For testing, the mixture 101 can be extruded within agap between two plates that are set to be approximately 8 mm apart fromeach other. After extruding the gel into the gap, the distance betweenthe two plates defining the gap is reduced to 2 mm until the mixture 101completely fills the gap between the plates. After wiping away excessmixture, the gap is decreased by 0.1 mm and the test is initiated. Thetest is an oscillation strain sweep test conducted with instrumentsettings of a strain range between 0.01% to 100%, at 6.28 rad/s (1 Hz),using 25-mm parallel plate and recording 10 points per decade. Within 1hour after the test completes, the gap is lowered again by 0.1 mm andthe test is repeated. The test can be repeated at least 6 times. Thefirst test may differ from the second and third tests. Only the resultsfrom the second and third tests for each specimen should be reported.

Furthermore, to facilitate processing and forming shaped abrasiveparticles according to embodiments herein, the mixture 101 can have aparticular viscosity. For example, the mixture 101 can have a viscosityof at least about 2×10³ Pa s, such as at least about 3×10³ Pa s, atleast about 4×10³ Pa s, at least about 5×10³ Pa s, at least about 6×10³Pa s, at least about 8×10³ Pa s, at least about 10×10³ Pa s, at leastabout 20×10³ Pa s, at least about 30×10³ Pa s, at least about 40×10³ Pas, at least about 50×10³ Pa s, at least about 60×10³ Pa s, or at leastabout 65×10³ Pa s. In at least one non-limiting embodiment, the mixture101 may have a viscosity of not greater than about 100×10³ Pa s, such asnot greater than about 95×10³ Pa s, not greater than about 90×10³ Pa s,or even not greater than about 85×10³ Pa s. It will be appreciated thatthe viscosity of the mixture 101 can be within a range between any ofthe minimum and maximum values noted above. The viscosity can bemeasured in the same manner as the storage modulus as described above.

Moreover, the mixture 101 can be formed to have a particular content oforganic materials including, for example, organic additives that can bedistinct from the liquid to facilitate processing and formation ofshaped abrasive particles according to the embodiments herein. Somesuitable organic additives can include stabilizers, binders such asfructose, sucrose, lactose, glucose, UV curable resins, and the like.

Notably, the embodiments herein may utilize a mixture 101 that can bedistinct from slurries used in conventional forming operations. Forexample, the content of organic materials within the mixture 101 and, inparticular, any of the organic additives noted above, may be a minoramount as compared to other components within the mixture 101. In atleast one embodiment, the mixture 101 can be formed to have not greaterthan about 30 wt % organic material for the total weight of the mixture101. In other instances, the amount of organic materials may be less,such as not greater than about 15 wt %, not greater than about 10 wt %,or even not greater than about 5 wt %. Still, in at least onenon-limiting embodiment, the amount of organic materials within themixture 101 can be at least about 0.01 wt %, such as at least about 0.5wt % for the total weight of the mixture 101. It will be appreciatedthat the amount of organic materials in the mixture 101 can be within arange between any of the minimum and maximum values noted above.

Moreover, the mixture 101 can be formed to have a particular content ofacid or base, distinct from the liquid content, to facilitate processingand formation of shaped abrasive particles according to the embodimentsherein. Some suitable acids or bases can include nitric acid, sulfuricacid, citric acid, chloric acid, tartaric acid, phosphoric acid,ammonium nitrate, and ammonium citrate. According to one particularembodiment in which a nitric acid additive is used, the mixture 101 canhave a pH of less than about 5, and more particularly, can have a pHwithin a range between about 2 and about 4.

The system 150 of FIG. 1 can include a die 103. As illustrated, themixture 101 can be provided within the interior of the die 103 andconfigured to be extruded through a die opening 105 positioned at oneend of the die 103. As further illustrated, extruding can includeapplying a force 180 on the mixture 101 to facilitate extruding themixture 101 through the die opening 105. During extrusion within anapplication zone 183, a tool 151 can be in direct contact with a portionof the die 103 and facilitate extrusion of the mixture 101 into the toolcavities 152. The tool 151 can be in the form of a screen, such asillustrated in FIG. 1, wherein the cavities 152 extend through theentire thickness of the tool 151. Still, it will be appreciated that thetool 151 may be formed such that the cavities 152 extend for a portionof the entire thickness of the tool 151 and have a bottom surface, suchthat the volume of space configured to hold and shape the mixture 101 isdefined by a bottom surface and side surfaces.

The tool 151 may be formed of a metal material, including for example, ametal alloy, such as stainless steel. In other instances, the tool 151may be formed of an organic material, such as a polymer.

In accordance with an embodiment, a particular pressure may be utilizedduring extrusion. For example, the pressure can be at least about 10kPa, such as at least about 500 kPa. Still, in at least one non-limitingembodiment, the pressure utilized during extrusion can be not greaterthan about 4 MPa. It will be appreciated that the pressure used toextrude the mixture 101 can be within a range between any of the minimumand maximum values noted above. In particular instances, the consistencyof the pressure delivered by a piston 199 may facilitate improvedprocessing and formation of shaped abrasive particles. Notably,controlled delivery of consistent pressure across the mixture 101 andacross the width of the die 103 can facilitate improved processingcontrol and improved dimensional characteristics of the shaped abrasiveparticles.

Prior to depositing the mixture 101 in the tool cavities 152, a moldrelease agent can be applied to the surfaces of the tool cavities 152,which may facilitate removal of precursor shaped abrasive particles fromthe tool cavities 152 after further processing. Such a process can beoptional and may not necessarily be used to conduct the molding process.A suitable exemplary mold release agent can include an organic material,such as one or more polymers (e.g., PTFE). In other instances, an oil(synthetic or organic) may be applied as a mold release agent to thesurfaces of the tool cavities 152. One suitable oil may be peanut oil.The mold release agent may be applied using any suitable manner,including but not limited to, depositing, spraying, printing, brushing,coating, and the like.

The mixture 101 may be deposited within the tool cavities 152, which maybe shaped in any suitable manner to form shaped abrasive particleshaving shapes corresponding to the shape of the tool cavities 152.

Referring briefly to FIG. 2, a portion of the tool 151 is illustrated.As shown, the tool 151 can include the tool cavities 152, and moreparticularly, a plurality of tool cavities 152 extending into the volumeof the tool 151. In accordance with an embodiment, the tool cavities 152can have a two-dimensional shape as viewed in a plane defined by thelength (l) and width (w) of the tool 151. The two-dimensional shape caninclude various shapes such as, for example, polygons, ellipsoids,numerals, Greek alphabet letters, Latin alphabet letters, Russianalphabet characters, complex shapes including a combination of polygonalshapes, and a combination thereof. In particular instances, the toolcavities 152 may have two-dimensional polygonal shapes such as arectangle, a quadrilateral, a pentagon, a hexagon, a heptagon, anoctagon, a nonagon, a decagon, and a combination thereof. Notably, aswill be appreciated in further reference to the shaped abrasiveparticles of the embodiments herein, the tool cavities 152 may utilizevarious other shapes.

While the tool 151 of FIG. 2 is illustrated as having tool cavities 152oriented in a particular manner relative to each other, it will beappreciated that various other orientations may be utilized. Inaccordance with one embodiment, each of the tool cavities 152 can havesubstantially the same orientation relative to each other, andsubstantially the same orientation relative to the surface of thescreen. For example, each of the tool cavities 152 can have a first edge154 defining a first plane 155 for a first row 156 of the tool cavities152 extending laterally across a lateral axis 158 of the tool 151. Thefirst plane 155 can extend in a direction substantially orthogonal to alongitudinal axis 157 of the tool 151. However, it will be appreciated,that in other instances, the tool cavities 152 need not necessarily havethe same orientation relative to each other.

Moreover, the first row 156 of tool cavities 152 can be orientedrelative to a direction of translation to facilitate particularprocessing and controlled formation of shaped abrasive particles. Forexample, the tool cavities 152 can be arranged on the tool 151 such thatthe first plane 155 of the first row 156 defines an angle relative tothe direction of translation 171. As illustrated, the first plane 155can define an angle that is substantially orthogonal to the direction oftranslation 171. Still, it will be appreciated that in one embodiment,the tool cavities 152 can be arranged on the tool 151 such that thefirst plane 155 of the first row 156 defines a different angle withrespect to the direction of translation, including for example, an acuteangle or an obtuse angle. Still, it will be appreciated that the toolcavities 152 may not necessarily be arranged in rows. The tool cavities152 may be arranged in various particular ordered distributions withrespect to each other on the tool 151, such as in the form of atwo-dimensional pattern. Alternatively, the openings may be disposed ina random manner on the tool 151.

Referring again to FIG. 1, during operation of the system 150, the tool151 can be translated in a direction 153 to facilitate a continuousmolding operation. As will be appreciated, the tool 151 may be in theform of a continuous belt, which can be translated over rollers tofacilitate continuous processing. In some embodiments, the tool 151 canbe translated while extruding the mixture 101 through the die opening105. As illustrated in the system 150, the mixture 101 may be extrudedin a direction 191. The direction of translation 153 of the tool 151 canbe angled relative to the direction of extrusion 191 of the mixture 101.While the angle between the direction of translation 153 and thedirection of extrusion 191 is illustrated as substantially orthogonal inthe system 100, other angles are contemplated, including for example, anacute angle or an obtuse angle. After the mixture 101 is extrudedthrough the die opening 105, the mixture 101 and tool 151 may betranslated under a knife edge 107 attached to a surface of the die 103.The knife edge 107 may define a region at the front of the die 103 thatfacilitates displacement of the mixture 101 into the tool cavities 152of the tool 151.

In the molding process, the mixture 101 may undergo significant dryingwhile contained in the tool cavity 152. Therefore, shaping may beprimarily attributed to substantial drying and solidification of themixture 101 in the tool cavities 152 to shape the mixture 101. Incertain instances, the shaped abrasive particles formed according to themolding process may exhibit shapes more closely replicating the featuresof the mold cavity compared to other processes, including for example,screen printing processes. However, it should be noted that certainbeneficial shape characteristics may be more readily achieved throughscreen printing processes.

After applying the mold release agent, the mixture 101 can be depositedwithin the mold cavities and dried. Drying may include removal of aparticular content of certain materials from the mixture 101, includingvolatiles, such as water or organic materials. In accordance with anembodiment, the drying process can be conducted at a drying temperatureof not greater than about 300° C., such as not greater than about 250°C., not greater than about 200° C., not greater than about 150° C., notgreater than about 100° C., not greater than about 80° C., not greaterthan about 60° C., not greater than about 40° C., or even not greaterthan about 30° C. Still, in one non-limiting embodiment, the dryingprocess may be conducted at a drying temperature of at least about −20°C., such as at least about −10° C. at least about 0° C. at least about5° C. at least about 10° C., or even at least about 20° C. It will beappreciated that the drying temperature may be within a range betweenany of the minimum and maximum temperatures noted above.

In certain instances, drying may be conducted for a particular durationto facilitate the formation of shaped abrasive particles according toembodiments herein. For example, drying can be conducted for a durationof at least about 1 minute, such as at least about 2 minutes, at leastabout 4 minutes, at least about 6 minutes, at least about 8 minutes, atleast about 10 minutes, at least about 30 minutes, at least about 1hour, at least about 2 hours, at least about 4 hours, at least about 8hours, at least about 12 hours, at least about 15 hours, at least about18 hours, at least about 24 hours. In still other instances, the processof drying may be not greater than about 30 hours, such as not greaterthan about 24 hours, not greater than about 20 hours, not greater thanabout 15 hours, not greater than about 12 hours, not greater than about10 hours, not greater than about 8 hours, not greater than about 6hours, not greater than about 4 hours. It will be appreciated that theduration of drying can be within a range between any of the minimum andmaximum values noted above.

Additionally, drying may be conducted at a particular relative humidityto facilitate formation of shaped abrasive particles according to theembodiments herein. For example, drying may be conducted at a relativehumidity of at least about 20%, at least about 30%, at least about 40%,at least about 50%, at least about 60%, such as at least about 62%, atleast about 64%, at least about 66%, at least about 68%, at least about70%, at least about 72%, at least about 74%, at least about 76%, atleast about 78%, or even at least about 80%. In still other non-limitingembodiments, drying may be conducted at a relative humidity of notgreater than about 90%, such as not greater than about 88%, not greaterthan about 86%, not greater than about 84%, not greater than about 82%,not greater than about 80%, not greater than about 78%, not greater thanabout 76%, not greater than about 74%, not greater than about 72%, notgreater than about 70%, not greater than about 65%, not greater thanabout 60%, not greater than about 55%, not greater than about 50%, notgreater than about 45%, not greater than about 40%, not greater thanabout 35%, not greater than about 30%, or even not greater than about25%. It will be appreciated that the relative humidity utilized duringdrying can be within a range between any of the minimum and maximumpercentages noted above.

After completing the drying process, the mixture 101 can be releasedfrom the tool cavities 152 to produce precursor shaped abrasiveparticles. Notably, before the mixture 101 is removed from the toolcavities 152 or after the mixture 101 is removed and the precursorshaped abrasive particles are formed, one or more post-forming processesmay be completed. Such processes can include surface shaping, curing,reacting, radiating, planarizing, calcining, sintering, sieving, doping,and a combination thereof. For example, in one optional process, themixture 101 or precursor shaped abrasive particles may be translatedthrough an optional shaping zone, wherein at least one exterior surfaceof the mixture or precursor shaped abrasive particles may be shaped. Instill another embodiment, the mixture 101 as contained in the moldcavities or the precursor shaped abrasive particles may be translatedthrough an optional application zone, wherein a dopant material can beapplied. In particular instances, the process of applying a dopantmaterial can include selective placement of the dopant material on atleast one exterior surface of the mixture 101 or precursor shapedabrasive particles.

The dopant material may be applied utilizing various methods includingfor example, spraying, dipping, depositing, impregnating, transferring,punching, cutting, pressing, crushing, and any combination thereof. Inaccordance with an embodiment, applying a dopant material can includethe application of a particular material, such as a precursor. Incertain instances, the precursor can be a salt, such as a metal salt,that includes a dopant material to be incorporated into thefinally-formed shaped abrasive particles. For example, the metal saltcan include an element or compound that is the precursor to the dopantmaterial. It will be appreciated that the salt material may be in liquidform, such as in a dispersion comprising the salt and liquid carrier.The salt may include nitrogen, and more particularly, can include anitrate. In other embodiments, the salt can be a chloride, sulfate,phosphate, and a combination thereof. In one embodiment, the salt caninclude a metal nitrate, and more particularly, consist essentially of ametal nitrate. In one embodiment, the dopant material can include anelement or compound such as an alkali element, alkaline earth element,rare earth element, hafnium, zirconium, niobium, tantalum, molybdenum,vanadium, or a combination thereof. In one particular embodiment, thedopant material includes an element or compound including an elementsuch as lithium, sodium, potassium, magnesium, calcium, strontium,barium, scandium, yttrium, lanthanum, cesium, praseodymium, niobium,hafnium, zirconium, tantalum, molybdenum, vanadium, chromium, cobalt,iron, germanium, manganese, nickel, titanium, zinc, and a combinationthereof.

The molding process may further include a sintering process. For certainembodiments herein, sintering can be conducted after removing themixture from the tool cavities 152 and forming the precursor shapedabrasive particles. Sintering of the precursor shaped abrasive particles123 may be utilized to densify the particles, which are generally in agreen state. In a particular instance, the sintering process canfacilitate the formation of a high-temperature phase of the ceramicmaterial. For example, in one embodiment, the precursor shaped abrasiveparticles may be sintered such that a high-temperature phase of alumina,such as alpha alumina, is formed. In one instance, a shaped abrasiveparticle can comprise at least about 90 wt % alpha alumina for the totalweight of the particle. In other instances, the content of alpha aluminamay be greater such that the shaped abrasive particle may consistessentially of alpha alumina.

The body of the finally-formed shaped abrasive particles can haveparticular two-dimensional shapes. For example, the body can have atwo-dimensional shape, as viewed in a plane defined by the length andwidth of the body, and can have a shape including a polygonal shape,ellipsoidal shape, a numeral, a Greek alphabet character, a Latinalphabet character, a Russian alphabet character, a complex shapeutilizing a combination of polygonal shapes and a combination thereof.Particular polygonal shapes include rectangular, trapezoidal,pentagonal, hexagonal, heptagonal, octagonal, nonagonal, decagonal, andany combination thereof. In another instance, the finally-formed shapedabrasive particles can have a body having a two-dimensional shape suchas an irregular quadrilateral, an irregular rectangle, an irregulartrapezoid, an irregular pentagon, an irregular hexagon, an irregularheptagon, an irregular octagon, an irregular nonagon, an irregulardecagon, and a combination thereof. An irregular polygonal shape is onewhere at least one of the sides defining the polygonal shape isdifferent in dimension (e.g., length) with respect to another side. Asillustrated in other embodiments herein, the two-dimensional shape ofcertain shaped abrasive particles can have a particular number ofexterior points or external corners. For example, the body of the shapedabrasive particles can have a two-dimensional polygonal shape as viewedin a plane defined by a length and width, wherein the body comprises atwo-dimensional shape having at least 4 exterior points (e.g., aquadrilateral), at least 5 exterior points (e.g., a pentagon), at least6 exterior points (e.g., a hexagon), at least 7 exterior points (e.g., aheptagon), at least 8 exterior points (e.g., an octagon), at least 9exterior points (e.g., a nonagon), and the like.

FIG. 3 includes a cross-sectional illustration of a shaped abrasiveparticle to illustrate certain features of shaped abrasive particles ofthe embodiments herein. It will be appreciated that such across-sectional view can be applied to any of the exemplary shapedabrasive particles of the embodiments to determine one or more shapeaspects or dimensional characteristics as described herein. The body ofthe shaped abrasive particle can include an upper major surface 303(i.e., a first major surface) and a bottom major surface 304 (i.e., asecond major surface) opposite the upper major surface 303. The uppersurface 303 and the bottom surface 304 can be separated from each otherby a side surface 314.

In certain instances, the shaped abrasive particles of the embodimentsherein, can have an average difference in height, which is a measure ofthe difference between hc and hm. Notably, the dimension of Lmiddle canbe a length defining a distance between a height at a corner (hc) and aheight at a midpoint edge (hm) opposite the corner. Moreover, the body301 can have an interior height (hi), which can be the smallestdimension of height of the body 301 as measured along a dimensionbetween any corner and opposite midpoint edge on the body 301. Forconvention herein, average difference in height will be generallyidentified as hc-hm, however it is defined as an absolute value of thedifference. Therefore, it will be appreciated that average difference inheight may be calculated as hm-hc when the height of the body 301 at theside surface 314 is greater than the height at the corner 313. Moreparticularly, the average difference in height can be calculated basedupon a plurality of shaped abrasive particles from a suitable samplesize. The heights hc and hm of the particles can be measured using aSTIL (Sciences et Techniques Industrielles de la Lumiere—France) MicroMeasure 3D Surface Profilometer (white light (LED) chromatic aberrationtechnique) and the average difference in height can be calculated basedon the average values of hc and hm from the sample.

As illustrated in FIG. 3, in one particular embodiment, the body 301 ofthe shaped abrasive particle 300 may have an average difference inheight at different locations at the body 301. The body 301 can have anaverage difference in height, which can be the absolute value of [hc−hm]between the first corner height (hc) and the second midpoint height (hm)that is quite low, such that the particle is relative flat, having anaverage difference in height that is not greater than about 300 microns,such as not greater than about 250 microns, not greater than about 220microns, not greater than about 180 microns, not greater than about 150microns, not greater than about 100 microns, not greater than about 50microns, or even not greater than about 20 microns.

The body of the shaped abrasive particles herein can include a width (w)that is the longest dimension of the body and extending along a side.The shaped abrasive particles can include a length that extends througha midpoint of the body and bisecting the body (i.e., Lmiddle). The bodycan further include a height (h), which may be a dimension of the bodyextending in a direction perpendicular to the length and width in adirection defined by a side surface of the body 301. In specificinstances, the width can be greater than or equal to the length, thelength can be greater than or equal to the height, and the width can begreater than or equal to the height.

In particular instances, the body 301 can be formed to have a primaryaspect ratio, which is a ratio expressed as width:length, having a valueof at least 1:1. In other instances, the body 301 can be formed suchthat the primary aspect ratio (w:1) is at least about 1.5:1, such as atleast about 2:1, at least about 4:1, or even at least about 5:1. Still,in other instances, the abrasive particle 300 can be formed such thatthe body 301 has a primary aspect ratio that is not greater than about10:1, such as not greater than 9:1, not greater than about 8:1, or evennot greater than about 5:1. It will be appreciated that the body 301 canhave a primary aspect ratio within a range between any of the ratiosnoted above. Furthermore, it will be appreciated that reference hereinto a height can be reference to the maximum height measurable of theabrasive particle 300.

In addition to the primary aspect ratio, the abrasive particle 300 canbe formed such that the body 301 comprises a secondary aspect ratio,which can be defined as a ratio of length:height, wherein the height isan interior median height (Mhi). In certain instances, the secondaryaspect ratio can be at least about 1:1, such as at least about 2:1, atleast about 4:1, or even at least about 5:1. Still, in other instances,the abrasive particle 300 can be formed such that the body 301 has asecondary aspect ratio that is not greater than about 1:3, such as notgreater than 1:2, or even not greater than about 1:1. It will beappreciated that the body 301 can have a secondary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 5:1 and about 1:1.

In accordance with another embodiment, the abrasive particle 300 can beformed such that the body 301 comprises a tertiary aspect ratio, definedby the ratio width:height, wherein the height is an interior medianheight (Mhi). The tertiary aspect ratio of the body 301 can be can be atleast about 1:1, such as at least about 2:1, at least about 4:1, atleast about 5:1, or even at least about 6:1. Still, in other instances,the abrasive particle 300 can be formed such that the body 301 has atertiary aspect ratio that is not greater than about 3:1, such as notgreater than 2:1, or even not greater than about 1:1. It will beappreciated that the body 301 can have a tertiary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 6:1 and about 1:1.

According to one embodiment, the body 301 of the shaped abrasiveparticle 300 can have particular dimensions, which may facilitateimproved performance. For example, in one instance, the body 301 canhave an interior height (hi), which can be the smallest dimension ofheight of the body 301 as measured along a dimension between any cornerand opposite midpoint edge on the body 301. In particular instances, theinterior height (hi) may be the smallest dimension of height (i.e.,measure between the bottom surface 304 and the upper surface 305) of thebody 301 for three measurements taken between each of the exteriorcorners and the opposite midpoint edges. The interior height (hi) of thebody 301 of a shaped abrasive particle 300 is illustrated in FIG. 3.According to one embodiment, the interior height (hi) can be at leastabout 20% of the width (w). The height (hi) may be measured bysectioning or mounting and grinding the shaped abrasive particle 300 andviewing in a manner sufficient (e.g., light microscope or SEM) todetermine the smallest height (hi) within the interior of the body 301.In one particular embodiment, the height (hi) can be at least about 22%of the width, such as at least about 25%, at least about 30%, or even atleast about 33%, of the width of the body 301. For one non-limitingembodiment, the height (hi) of the body 301 can be not greater thanabout 80% of the width of the body 301, such as not greater than about76%, not greater than about 73%, not greater than about 70%, not greaterthan about 68% of the width, not greater than about 56% of the width,not greater than about 48% of the width, or even not greater than about40% of the width. It will be appreciated that the height (hi) of thebody 301 can be within a range between any of the above noted minimumand maximum percentages.

A batch of shaped abrasive particles can be fabricated where the medianinterior height value (Mhi) can be controlled, which may facilitateimproved performance. In particular, the median internal height (hi) ofa batch can be related to a median width of the shaped abrasiveparticles of the batch in the same manner as described above. Notably,the median interior height (Mhi) can be at least about 20% of the width,such as at least about 22%, at least about 25%, at least about 30%, oreven at least about 33% of the median width of the shaped abrasiveparticles of the batch. For one non-limiting embodiment, the medianinterior height (Mhi) of the body 301 can be not greater than about 80%,such as not greater than about 76%, not greater than about 73%, notgreater than about 70%, not greater than about 68% of the width, notgreater than about 56% of the width, not greater than about 48% of thewidth, or even not greater than about 40% of the median width of thebody 301. It will be appreciated that the median interior height (Mhi)of the body 301 can be within a range between any of the above notedminimum and maximum percentages.

Furthermore, the batch of shaped abrasive particles may exhibit improveddimensional uniformity as measured by the standard deviation of adimensional characteristic from a suitable sample size. According to oneembodiment, the shaped abrasive particles can have an interior heightvariation (Vhi), which can be calculated as the standard deviation ofinterior height (hi) for a suitable sample size of particles from abatch. According to one embodiment, the interior height variation can benot greater than about 60 microns, such as not greater than about 58microns, not greater than about 56 microns, or even not greater thanabout 54 microns. In one non-limiting embodiment, the interior heightvariation (Vhi) can be at least about 2 microns. It will be appreciatedthat the interior height variation of the body can be within a rangebetween any of the above noted minimum and maximum values.

For another embodiment, the body 301 of the shaped abrasive particle 300can have a height, which may be an interior height (hi), of at leastabout 70 microns. More particularly, the height may be at least about 80microns, such as at least about 90 microns, at least about 100 microns,at least about 110 microns, at least about 120 microns, at least about150 microns, at least about 175 microns, at least about 200 microns, atleast about 225 microns, at least about 250 microns, at least about 275microns, or even at least about 300 microns. In still one non-limitingembodiment, the height of the body 301 can be not greater than about 3mm, such as not greater than about 2 mm, not greater than about 1.5 mm,not greater than about 1 mm, or even not greater than about 800 microns,not greater than about 600 microns, not greater than about 500 microns,not greater than about 475 microns, not greater than about 450 microns,not greater than about 425 microns, not greater than about 400 microns,not greater than about 375 microns, not greater than about 350 microns,not greater than about 325 microns, not greater than about 300 microns,not greater than about 275 microns, or even not greater than about 250microns. It will be appreciated that the height of the body 301 can bewithin a range between any of the above noted minimum and maximumvalues. Moreover, it will be appreciated that the above range of valuescan be representative of a median interior height (Mhi) value for abatch of shaped abrasive particles.

For certain embodiments herein, the body 301 of the shaped abrasiveparticle 300 can have particular dimensions, including for example, awidth≥length, a length≥height, and a width≥height. More particularly,the body 301 of the shaped abrasive particle 300 can have a width (w) ofat least about 200 microns, such as at least about 250 microns, at leastabout 300 microns, at least about 350 microns, at least about 400microns, at least about 450 microns, at least about 500 microns, atleast about 550 microns, at least about 600 microns, at least about 700microns, at least about 800 microns, or even at least about 900 microns.In one non-limiting instance, the body 301 can have a width of notgreater than about 4 mm, such as not greater than about 3 mm, notgreater than about 2.5 mm, or even not greater than about 2 mm. It willbe appreciated that the width of the body 301 can be within a rangebetween any of the above noted minimum and maximum values. Moreover, itwill be appreciated that the above range of values can be representativeof a median width (Mw) for a batch of shaped abrasive particles.

The body 301 of the shaped abrasive particle 300 can have particulardimensions, including for example, a length (Lmiddle or Lp) of at leastabout 0.4 mm, such as at least about 0.6 mm, at least about 0.8 mm, oreven at least about 0.9 mm. Still, for at least one non-limitingembodiment, the body 301 can have a length of not greater than about 4mm, such as not greater than about 3 mm, not greater than about 2.5 mm,or even not greater than about 2 mm. It will be appreciated that thelength of the body 301 can be within a range between any of the abovenoted minimum and maximum values. Moreover, it will be appreciated thatthe above range of values can be representative of a median length (Ml),which may be more particularly a median middle length (MLmiddle) ormedian profile length (MLp), for a batch of shaped abrasive particles.

The shaped abrasive particle 300 can have a body 301 having a particularamount of dishing, wherein the dishing value (d) can be defined as aratio between an average height of the body 301 at the exterior corners(Ahc) as compared to the smallest dimension of height of the body 301 atthe interior (hi). The average height of the body 301 at the corners(Ahc) can be calculated by measuring the height of the body 301 at allcorners and averaging the values, and may be distinct from a singlevalue of height at one corner (hc). The average height of the body 301at the corners or at the interior can be measured using a STIL (Scienceset Techniques Industrielles de la Lumiere—France) Micro Measure 3DSurface Profilometer (white light (LED) chromatic aberration technique).Alternatively, the dishing may be based upon a median height of theparticles at the corner (Mhc) calculated from a suitable sampling ofparticles from a batch. Likewise, the interior height (hi) can be amedian interior height (Mhi) derived from a suitable sampling of shapedabrasive particles from a batch. According to one embodiment, thedishing value (d) can be not greater than about 2, such as not greaterthan about 1.9, not greater than about 1.8, not greater than about 1.7,not greater than about 1.6, not greater than about 1.5, or even notgreater than about 1.2. Still, in at least one non-limiting embodiment,the dishing value (d) can be at least about 0.9, such as at least about1.0. It will be appreciated that the dishing ratio can be within a rangebetween any of the minimum and maximum values noted above. Moreover, itwill be appreciated that the above dishing values can be representativeof a median dishing value (Md) for a batch of shaped abrasive particles.

The shaped abrasive particles of the embodiments herein, including forexample, the body 301 of the particle of FIG. 3 can have a bottomsurface 304 defining a bottom area (A_(b)). In particular instances, thebottom surface 304 can be the largest surface of the body 301. Thebottom major surface 304 can have a surface area defined as the bottomarea (A_(b)) that is different than the surface area of the upper majorsurface 303. In one particular embodiment, the bottom major surface 304can have a surface area defined as the bottom area (A_(b)) that isdifferent than the surface area of the upper major surface 303. Inanother embodiment, the bottom major surface 304 can have a surface areadefined as the bottom area (A_(b)) that is less than the surface area ofthe upper major surface 303.

Additionally, the body 301 can have a cross-sectional midpoint area(A_(m)) defining an area of a plane perpendicular to the bottom area(A_(b)) and extending through a midpoint 381 of the particle 300. Incertain instances, the body 301 can have an area ratio of bottom area tomidpoint area (A_(b)/A_(m)) of not greater than about 6. In moreparticular instances, the area ratio can be not greater than about 5.5,such as not greater than about 5, not greater than about 4.5, notgreater than about 4, not greater than about 3.5, or even not greaterthan about 3. Still, in one non-limiting embodiment, the area ratio maybe at least about 1.1, such as at least about 1.3, or even at leastabout 1.8. It will be appreciated that the area ratio can be within arange between any of the minimum and maximum values noted above.Moreover, it will be appreciated that the above area ratios can berepresentative of a median area ratio for a batch of shaped abrasiveparticles.

Furthermore the shaped abrasive particles of the embodiments hereinincluding, for example, the particle of FIG. 3, can have a normalizedheight difference of not greater than about 0.3. The normalized heightdifference can be defined by the absolute value of the equation[(hc−hm)/(hi)]. In other embodiments, the normalized height differencecan be not greater than about 0.26, such as not greater than about 0.22,or even not greater than about 0.19. Still, in one particularembodiment, the normalized height difference can be at least about 0.04,such as at least about 0.05, or even at least about 0.06. It will beappreciated that the normalized height difference can be within a rangebetween any of the minimum and maximum values noted above. Moreover, itwill be appreciated that the above normalized height values can berepresentative of a median normalized height value for a batch of shapedabrasive particles.

The shaped abrasive particle 300 can be formed such that the body 301includes a crystalline material, and more particularly, apolycrystalline material. Notably, the polycrystalline material caninclude abrasive grains. In one embodiment, the body 301 can beessentially free of an organic material, including for example, abinder. More particularly, the body 301 can consist essentially of apolycrystalline material.

In one aspect, the body 301 of the shaped abrasive particle 300 can bean agglomerate including a plurality of abrasive particles, grit, and/orgrains bonded to each other to form the body 301 of the abrasiveparticle 300. Suitable abrasive grains can include nitrides, oxides,carbides, borides, oxynitrides, oxyborides, diamond, and a combinationthereof. In particular instances, the abrasive grains can include anoxide compound or complex, such as aluminum oxide, zirconium oxide,titanium oxide, yttrium oxide, chromium oxide, strontium oxide, siliconoxide, and a combination thereof. In one particular instance, theabrasive particle 300 is formed such that the abrasive grains formingthe body 301 include alumina, and more particularly, may consistessentially of alumina. Moreover, in particular instances, the shapedabrasive particle 300 can be formed from a seeded sol-gel.

The abrasive grains (i.e., crystallites) contained within the body 301may have an average grain size that is generally not greater than about100 microns. In other embodiments, the average grain size can be less,such as not greater than about 80 microns, not greater than about 50microns, not greater than about 30 microns, not greater than about 20microns, not greater than about 10 microns, or even not greater thanabout 1 micron, not greater than about 0.9 microns, not greater thanabout 0.8 microns, not greater than about 0.7 microns, or even notgreater than about 0.6 microns. Still, the average grain size of theabrasive grains contained within the body 301 can be at least about 0.01microns, such as at least about 0.05 microns, at least about 0.06microns, at least about 0.07 microns, at least about 0.08 microns, atleast about 0.09 microns, at least about 0.1 microns, at least about0.12 microns, at least about 0.15 microns, at least about 0.17 microns,at least about 0.2 microns, or even at least about 0.5 microns. It willbe appreciated that the abrasive grains can have an average grain sizewithin a range between any of the minimum and maximum values notedabove.

In accordance with certain embodiments, the abrasive particle 300 can bea composite article including at least two different types of grainswithin the body 301. It will be appreciated that different types ofgrains are grains having different compositions with regard to eachother. For example, the body 301 can be formed such that is includes atleast two different types of grains, wherein the two different types ofgrains can be nitrides, oxides, carbides, borides, oxynitrides,oxyborides, diamond, and a combination thereof.

In accordance with an embodiment, the abrasive particle 300 can have anaverage particle size, as measured by the largest dimension measurableon the body 301, of at least about 100 microns. In fact, the abrasiveparticle 300 can have an average particle size of at least about 150microns, such as at least about 200 microns, at least about 300 microns,at least about 400 microns, at least about 500 microns, at least about600 microns, at least about 700 microns, at least about 800 microns, oreven at least about 900 microns. Still, the abrasive particle 300 canhave an average particle size that is not greater than about 5 mm, suchas not greater than about 3 mm, not greater than about 2 mm, or even notgreater than about 1.5 mm. It will be appreciated that the abrasiveparticle 300 can have an average particle size within a range betweenany of the minimum and maximum values noted above.

The shaped abrasive particles of the embodiments herein can have apercent flashing that may facilitate improved performance. Notably, theflashing defines an area of the particle as viewed along one side, suchas illustrated in FIG. 4, wherein the flashing extends from a sidesurface of the body 301 within the boxes 402 and 403. The flashing canrepresent tapered regions proximate to the upper surface 303 and bottomsurface 304 of the body 301. The flashing can be measured as thepercentage of area of the body 301 along the side surface containedwithin a box extending between an innermost point of the side surface(e.g., 421) and an outermost point (e.g., 422) on the side surface ofthe body 301. In one particular instance, the body 301 can have aparticular content of flashing, which can be the percentage of area ofthe body 301 contained within the boxes 402 and 403 compared to thetotal area of the body 301 contained within boxes 402, 403, and 404.According to one embodiment, the percent flashing (f) of the body 301can be at least about 1%. In another embodiment, the percent flashingcan be greater, such as at least about 2%, at least about 3%, at leastabout 5%, at least about 8%, at least about 10%, at least about 12%,such as at least about 15%, at least about 18%, or even at least about20%. Still, in a non-limiting embodiment, the percent flashing of thebody 301 can be controlled and may be not greater than about 45%, suchas not greater than about 40%, not greater than about 35%, not greaterthan about 30%, not greater than about 25%, not greater than about 20%,not greater than about 18%, not greater than about 15%, not greater thanabout 12%, not greater than about 10%, not greater than about 8%, notgreater than about 6%, or even not greater than about 4%. It will beappreciated that the percent flashing of the body 301 can be within arange between any of the above minimum and maximum percentages.Moreover, it will be appreciated that the above flashing percentages canbe representative of an average flashing percentage or a median flashingpercentage for a batch of shaped abrasive particles.

The percent flashing can be measured by mounting the shaped abrasiveparticle 300 on its side and viewing the body 301 at the side togenerate a black and white image, such as illustrated in FIG. 4. Asuitable program for such includes ImageJ software. The percentageflashing can be calculated by determining the area of the body 301 inthe boxes 402 and 403 compared to the total area of the body 301 asviewed at the side (total shaded area), including the area in the center404 and within the boxes. Such a procedure can be completed for asuitable sampling of particles to generate average, median, and/or andstandard deviation values.

FIG. 12A includes a perspective view illustration of a shaped abrasiveparticle according to an embodiment. FIG. 12B includes a top viewillustration of a shaped abrasive particle according to an embodiment.As illustrated, the shaped abrasive particle 1200 can include a body1201 having an upper major surface 1203 (i.e., a first major surface)and a bottom major surface 1204 (i.e., a second major surface) oppositethe upper major surface 1203. The upper surface 1203 and the bottomsurface 1204 can be separated from each other by at least one sidesurface 1205, which may include one or more discrete side surfaceportions, including for example, a first portion 1206 of the sidesurface 1205, a second portion 1207 of the side surface 1205, and athird portion 1208 of the side surface 1205. In particular, the firstportion 1206 of the side surface 1205 can extend between a first corner1209 and a second corner 1210. Notably, the first corner 1209 can be anexternal corner joining two portions of the side surface 1205. The firstcorner 1209 and second corner 1210, which is also an external corner,are adjacent to each other and have no other external corners disposedbetween them. External corners of a body are defined by the joining oftwo linear sections when viewing the body of the shaped abrasiveparticle top down.

The second portion 1207 of the side surface 1205 can extend between asecond corner 1210 and a third corner 1211. Notably, the second corner1210 is an external corner joining two portions of the side surface1205. The second corner 1210 and third corner 1211, which is also anexternal corner, are adjacent to each other and have no other externalcorners disposed between them. Also, the third portion 1208 of the sidesurface 1205 can extend between the third corner 1211 and the firstcorner 1209, which are both external corners that are adjacent to eachother, having no other external corners disposed between them.

Moreover, as illustrated in the perspective view of FIG. 12A, the firstportion 1206, second portion 1207, and third portion 1208 of the sidesurface 1205 may be joined to each other by edges 1221, 1222, and 1223,respectively. The edges 1221, 1222, and 1223 extend between the uppermajor surface 1203 and the bottom major surface 1204.

The edge 1223 can extend between an external corner 1211 of the uppermajor surface 1203 and an external corner 1212 of the bottom majorsurface 1204. The edge 1222 can extend between an external corner 1210of the upper major surface 1203 and an external corner 1214 of thebottom major surface 1204. The edge 1221 can extend between an externalcorner 1209 of the upper major surface 1203 and an external corner 1215of the bottom major surface 1204.

The body 1201 can have a length (L or Lmiddle) as shown in FIG. 12A,which may be measured as the longest dimension extending from anexternal corner (e.g., 1210) to a midpoint at the opposite side surface(e.g., the third portion 1208 of the side surface 1205). Notably, insome embodiments, such as illustrated in FIG. 12A, the length can extendthrough a midpoint 1281 of the upper surface 1203 of the body 1201,however, this may not necessarily be the case for every embodiment.Moreover, the body 1201 can have a width (W), which is the measure ofthe longest dimension of the body 1201 along a discrete side surfaceportion of the side surface 1205. The height of the body may begenerally the distance between the upper major surface 1203 and thebottom major surface 1204. As described in embodiments herein, theheight may vary in dimension at different locations of the body 1201,such as at the corners versus at the interior of the body 1201.

As illustrated, the body 1201 of the shaped abrasive particle 1200 canhave a generally polygonal shape as viewed in a plane parallel to theupper surface 1203, and more particularly, a hybrid polygonaltwo-dimensional shape as viewed in the plane of the width and length ofthe body (i.e., the top view as shown in FIG. 12B), having 5 externalpoints or external corners.

As noted in other embodiments herein, the body 1201 can be formed tohave a primary aspect ratio, which can be a ratio expressed aswidth:length, having the values described in embodiments herein. Inother instances, the body 1201 can be formed such that the primaryaspect ratio (w:1) can be at least about 1.5:1, such as at least about2:1, at least about 4:1, or even at least about 5:1. Still, in otherinstances, the abrasive particle 1200 can be formed such that the body1201 has a primary aspect ratio that is not greater than about 10:1,such as not greater than 9:1, not greater than about 8:1, or even notgreater than about 5:1. It will be appreciated that the body 1201 canhave a primary aspect ratio within a range between any of the ratiosnoted above.

In addition to the primary aspect ratio, the abrasive particle 1200 canbe formed such that the body 1201 comprises a secondary aspect ratio,which can be defined as a ratio of length:height, wherein the height maybe an interior median height (Mhi) measured at the midpoint 1281. Incertain instances, the secondary aspect ratio can be at least about 1:1,such as at least about 2:1, at least about 4:1, or even at least about5:1. Still, in other instances, the abrasive particle 1200 can be formedsuch that the body 1201 has a secondary aspect ratio that is not greaterthan about 1:3, such as not greater than 1:2, or even not greater thanabout 1:1. It will be appreciated that the body 1201 can have asecondary aspect ratio within a range between any of the ratios notedabove, such as within a range between about 5:1 and about 1:1.

In accordance with another embodiment, the abrasive particle 1200 can beformed such that the body 1201 comprises a tertiary aspect ratio,defined by the ratio width:height, wherein the height may be an interiormedian height (Mhi). The tertiary aspect ratio of the body 1201 can beat least about 1:1, such as at least about 2:1, at least about 4:1, atleast about 5:1, or even at least about 6:1. Still, in other instances,the abrasive particle 1200 can be formed such that the body 1201 has atertiary aspect ratio that is not greater than about 3:1, such as notgreater than 2:1, or even not greater than about 1:1. It will beappreciated that the body 1201 can have a tertiary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 6:1 and about 1:1.

In one aspect, the body 1201 of the shaped abrasive particle 1200 canhave a first portion 1206 of the side surface 1205 with apartially-concave shape. As shown in FIG. 12B, a partially concave shapeincludes a curved section 1242 whose first curved section length (Lc1extends for a fraction of the total length (Lfp1) of the first portion1206 of the side surface 1205 between the adjacent corners 1209 and1210. In an embodiment, the total length (Lfp1) can be equivalents to awidth of the body 1201. Moreover, as further illustrated in theembodiment of FIGS. 12A and 12B, the first curved section 1242 can bedisposed between a first linear section 1241 and a second linear section1243. The first linear section 1241 can terminate at a first end at theexternal corner 1209 of the body 1201, extend along the first portion1206 of the side surface 1205 for a length (Ll1), and terminate at asecond end at the joining of the first linear section 1241 with thefirst curved section 1242. The first curved section 1242 and the firstlinear section 1241 can define a first interior corner 1245, which alongwith the first linear section 1241 and the first curved section 1242 candefine a first interior angle 1247 having an obtuse value. The secondlinear section 1243 can terminate at a first end at the external corner1210, extend along the first portion 1206 of the side surface 1205 for alength (Ll2), and terminate at a second end at the joining of the secondlinear section 1243 with the first curved section 1242. The secondlinear section 1243 and the first curved section 1242 can define asecond interior corner 1246. The second interior corner 1246, along withthe second linear section 1243 and the first curved section 1242 candefine a second interior angle 1248 having an obtuse value.

As will be appreciated, the first linear section 1241 and the secondlinear section 1243 can be substantially linear when viewed from the topdown, as illustrated in FIG. 12B. The first curved section 1242 can havea significant arcuate contour when viewed from the top down, also asshown in FIG. 12B. In certain instances, the body 1201 may be referredto as a hybrid polygonal shape, wherein a sum of the external corners issubstantially 180 degrees, and wherein at least a portion of the sidesurface (e.g., the first portion 1206) has an arcuate curvature, such asthe contour of the first curved section 1242.

As illustrated in FIG. 12B, the first linear section 1241 can have afirst linear section length (Ll1) and the first curved section 1242 canhave a first curved section length (Lc1). In certain embodiments, thelength of the first curved section 1242 can be not less than the lengthof the first linear section 1241 (i.e., Lc1≥Ll1). Still, in at least onenon-limiting embodiment, the length of the first linear section 1241 canbe not less than the length of the first curved section 1242 (i.e.,Ll1≥Lc1). In at least one particular instance, the relationship betweenthe length of the first linear section 1241 and the first curved section1242 may define a length factor (Ll1/Lc1) that may facilitate certainperformance of the shaped abrasive particle 1200. For example, thelength factor (Ll1/Lc1) can be not greater than about 1, such as notgreater than about 0.95, not greater than about 0.9, not greater thanabout 0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05. For yet another non-limitingembodiment, the length factor (Ll1/Lc1) can be at least about 0.05, suchas at least about 0.1, at least about 0.15, or even at least about 0.2.It will be appreciated that the length factor (Ll1/Lc1) can be within arange between any of the minimum and maximum values noted above.

In at least one alternative embodiment, the body 1201 can define anotherlength factor (Lc1/Ll1), which may be suitable for facilitating improvedperformance e of the shaped abrasive particle and having a value notgreater than about 1, such as not greater than about 0.95, not greaterthan about 0.9, not greater than about 0.85, not greater than about 0.8,not greater than about 0.75, not greater than about 0.7, not greaterthan about 0.65, not greater than about 0.6, not greater than about0.55, not greater than about 0.5, not greater than about 0.45, notgreater than about 0.4, not great not greater than about 0.35, notgreater than about 0.3, not greater than about 0.35, not greater thanabout 0.3, not greater than about 0.25, not greater than about 0.2, notgreater than about 0.15, not greater than about 0.1, or even not greaterthan about 0.05. In yet another embodiment, the length factor (Lc1/Ll1)can be at least about 0.05, such as at least about 0.1, at least about0.15, or even at least about 0.2. It will be appreciated that the lengthfactor (Lc1/Ll1) can be within a range between any of the minimum andmaximum values noted above.

As further illustrated, the second linear section 1243 can have a length(Ll2). In at least one embodiment, Ll1 and Ll2 can be substantiallyequal to each other. In still other instances, Ll1 and Ll2 can bemeasurably different compared to each other.

In another aspect, the second linear section 1243 can have a particularlength relative to the length of the first curved section 1242, whichmay facilitate improved performance of the body 1201. For example, inone embodiment, Lc1 can be not less than Ll2 (i.e., Lc1≥Ll2). In a moreparticular embodiment, the relationship between the length (Ll2) of thesecond linear section 1243 and the length (Lc1) of the first curvedsection 1242 can define a length factor (Ll2/Lc1), which may be notgreater than about 1, such as not greater than about 0.95, not greaterthan about 0.9, not greater than about 0.85, not greater than about 0.8,not greater than about 0.75, not greater than about 0.7, not greaterthan about 0.65, not greater than about 0.6, not greater than about0.55, not greater than about 0.5, not greater than about 0.45, notgreater than about 0.4, not great not greater than about 0.35, notgreater than about 0.3, not greater than about 0.35, not greater thanabout 0.3, not greater than about 0.25, not greater than about 0.2, notgreater than about 0.15, not greater than about 0.1, not greater thanabout 0.05. Still, in another non-limiting embodiment, the length factor(Ll2/Lc1) may be at least about 0.05, such as at least about 0.1, atleast about 0.15, or even at least about 0.2. It will be appreciatedthat the length factor (Ll2/Lc1) can be within a range between any ofthe minimum and maximum values noted above.

In still another embodiment, the relationship between the length (Ll2)of the second linear section 1243 and the length (Lc1) of the firstcurved section 1242 can define another length factor (Lc1/Ll2), whichmay be not greater than about 1, such as not greater than about 0.95,not greater than about 0.9, not greater than about 0.85, not greaterthan about 0.8, not greater than about 0.75, not greater than about 0.7,not greater than about 0.65, not greater than about 0.6, not greaterthan about 0.55, not greater than about 0.5, not greater than about0.45, not greater than about 0.4, not great not greater than about 0.35,not greater than about 0.3, not greater than about 0.35, not greaterthan about 0.3, not greater than about 0.25, not greater than about 0.2,not greater than about 0.15, not greater than about 0.1, not greaterthan about 0.05. In still another non-limiting embodiment, the lengthfactor (Lc1/Ll2) can be at least about 0.05, such as at least about 0.1,at least about 0.15, at least about 0.2. It will be appreciated that thelength factor (Lc1/Ll2) can be within a range between any of the minimumand maximum values noted above.

The body 1201 may be formed such that the first portion 1206 of the sidesurface 1205 has a particular relationship between the sum of the length(Ll1) of the first linear section 1241 and the length (Ll2) of thesecond linear section 1243, relative to the length (Lc1) of the firstcurved section 1242, such that a linear sum factor ((Ll1+Ll2)/Lc1) maybe controlled to facilitate improved performance of the body 1201.According to at least one embodiment, the linear sum factor can be notgreater than about 1, such as not greater than about 0.95, not greaterthan about 0.9, not greater than about 0.85, not greater than about 0.8,not greater than about 0.75, not greater than about 0.7, not greaterthan about 0.65, not greater than about 0.6, not greater than about0.55, not greater than about 0.5, not greater than about 0.45, notgreater than about 0.4, not great not greater than about 0.35, notgreater than about 0.3, not greater than about 0.35, not greater thanabout 0.3, not greater than about 0.25, not greater than about 0.2, notgreater than about 0.15, not greater than about 0.1, or even not greaterthan about 0.05. In yet another non-limiting embodiment, the linear sumfactor ((Ll1+Ll2)/Lc1) can be at least about 0.05, such as at leastabout 0.1, at least about 0.15, or even at least about 0.2. It will beappreciated that the linear sum factor ((Ll1+Ll2)/Lc1) can be within arange between any of the minimum and maximum values noted above.

For still another embodiment, the body 1201 may be formed such that thefirst portion 1206 of the side surface 1205 can have a particularrelationship between the sum of the length (Ll1) of the first linearsection 1241 and the length (Ll2) of the second linear section 1243,relative to the length (Lc1) of the first curved section 1242, such thatan inverse linear sum factor nc1/(Ll1+Ll2) is defined. The inverselinear sum factor can be controlled to facilitate improved performanceof the body 1201. In at least one embodiment the inverse linear sumfactor (Lc1/(Ll1+Ll2)) can be not greater than about 1, such as notgreater than about 0.95, not greater than about 0.9, not greater thanabout 0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, or even not greater than about 0.05. In yet anotherembodiment, the inverse linear sum factor (Lc1/(Ll1+Ll2)) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, oreven at least about 0.2. It will be appreciated that the inverse linearsum factor (Lc1/(Ll1+Ll2)) can be within a range between any of theminimum and maximum values noted above.

According to one embodiment, the first curved section 1242 can have aparticular first curved section length (Lc1) relative to the totallength (Lfp1) of the first portion 1206 that may facilitate improvedperformance of the body 1201. The total length (Lfp1) of the firstportion 1206 can be equivalent to a width (W) of the body 1201. Incertain instances, the first curved section length (Lc1) can be afraction of a total length (Lfp1) of the first portion 1206 of the sidesurface 1205. For example, the relationship between the first curvedsection length (Lc1) and the total length (Lfp1) of the first portion1206 can define a length factor (Lc1/Lfp1), which maybe not greater thanabout 1, such as not greater than about 0.95, not greater than about0.9, not greater than about 0.85, not greater than about 0.8, notgreater than about 0.75, not greater than about 0.7, not greater thanabout 0.65, not greater than about 0.6, not greater than about 0.55, notgreater than about 0.5, not greater than about 0.45, not greater thanabout 0.4, not great not greater than about 0.35, not greater than about0.3, not greater than about 0.35, not greater than about 0.3, notgreater than about 0.25, not greater than about 0.2, not greater thanabout 0.15, not greater than about 0.1, not greater than about 0.05.Still, in another non-limiting embodiment, the length factor (Lc1/Lfp1)may be at least about 0.05, such as at least about 0.1, at least about0.15, or even at least about 0.2. It will be appreciated that the lengthfactor (Lc1/Lfp1) can be within a range between any of the minimum andmaximum values noted above.

Further to the body 1201, the first linear section 1241 can have aparticular length (Ll1) relative to the total length (Lfp1) of the firstportion 1206 that may facilitate improved performance of the body 1201.In certain instances, the first linear section length (Ll1) can be afraction of a total length (Lfp1) of the first portion 1206 of the sidesurface 1205. For example, the relationship between the first linearsection length (Ll1) and the total length (Lfp1) of the first portion1206 can define a length factor (Ll1/Lfp1), which maybe not greater thanabout 1, such as not greater than about 0.95, not greater than about0.9, not greater than about 0.85, not greater than about 0.8, notgreater than about 0.75, not greater than about 0.7, not greater thanabout 0.65, not greater than about 0.6, not greater than about 0.55, notgreater than about 0.5, not greater than about 0.45, not greater thanabout 0.4, not great not greater than about 0.35, not greater than about0.3, not greater than about 0.35, not greater than about 0.3, notgreater than about 0.25, not greater than about 0.2, not greater thanabout 0.15, not greater than about 0.1, not greater than about 0.05.Still, in another non-limiting embodiment, the length factor (Ll1/Lfp1)may be at least about 0.05, such as at least about 0.1, at least about0.15, or even at least about 0.2. It will be appreciated that the lengthfactor (Ll1/Lfp1) can be within a range between any of the minimum andmaximum values noted above.

Moreover, the second linear section 1243 can have a particular length(Ll2) relative to the total length (Lfp1) of the first portion 1206 thatmay facilitate improved performance of the body 1201. In certaininstances, the second linear section length (Ll2) can be a fraction of atotal length (Lfp1) of the first portion 1206 of the side surface 1205.For example, the relationship between the second linear section length(Ll2) and the total length (Lfp1) of the first portion 1206 can define alength factor (Ll2/Lfp1), which maybe not greater than about 1, such asnot greater than about 0.95, not greater than about 0.9, not greaterthan about 0.85, not greater than about 0.8, not greater than about0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, not greater than about 0.05. Still, in anothernon-limiting embodiment, the length factor (Ll2/Lfp1) may be at leastabout 0.05, such as at least about 0.1, at least about 0.15, or even atleast about 0.2. It will be appreciated that the length factor(Ll2/Lfp1) can be within a range between any of the minimum and maximumvalues noted above.

As noted herein, and as illustrated in the embodiments of FIGS. 12A and12B, the first curved section 1242 can be joined to the first linearsection 1241 and define an interior corner 1245. Moreover, the firstcurved section 1242 can be joined to the second linear section 1243 anddefine an interior corner 1246. In particular instances, the firstcurved section 1242 can have a first end defined at the joint of theinterior corner 1245 that is spaced apart from the first external corner1209 of the body 1201. Moreover, the first curved section 1242 can havea second end defined at the joint of the interior corner 1246, which canbe spaced apart from the second external corner 1210 of the body 1201.Notably, in certain embodiments, the first portion 1206 of the sidesurface 1205 can include the first interior corner 1245 and the secondinterior corner 1246, which can be spaced apart from each other. Inparticular, the first interior corner 1245 and the second interiorcorner 1246 can be separated by the first curved section 1242, and moreparticularly, disposed at opposite ends of the first curved section1242. The first interior corner 1245 can be disposed at an edge betweenthe first linear section 1241 and the first curved section 1242 and thesecond interior corner 1246 can be disposed at an edge between the firstcurved section 1242 and the second linear section 1243.

The first interior corner 1245, along with the first curved section 1242and the first linear section 1241, can define the first interior angle1247, which can have an obtuse value. The first interior angle 1247 canbe measured as the angle formed between the first linear section 1241and a tangent 1283 of the first curved section 1242 that extends fromthe first interior corner 1245. According to one embodiment, the firstinterior angle 1247 can have a value between at least about 92 degreesand not greater than about 178 degrees. More particularly, in at leastone embodiment, the first interior angle 1247 can have a value of atleast about 94 degrees, such as at least about 96 degrees, at leastabout 98 degrees, at least about 100 degrees, at least about 102degrees, at least about 104 degrees, at least about 106 degrees, atleast about 108 degrees, at least about 110 degrees, at least about 112degrees, at least about 124 degrees, at least about 126 degrees, atleast about 128 degrees, at least about 120 degrees, at least about 122degrees, at least about 124 degrees, at least about 126 degrees, atleast about 128 degrees, at least about 130 degrees, at least about 132degrees, at least about 134 degrees, at least about 136 degrees, atleast about 138 degrees, or even at least about 140 degrees. In yetanother embodiment, the first interior angle 1247 can have a value ofnot greater than about 176 degrees, such as not greater than about 174degrees, not greater than about 172 degrees, not greater than about 170degrees, not greater than about 168 degrees, not greater than about 166degrees, not greater than about 164 degrees, not greater than about 162degrees, not greater than about 160 degrees, not greater than about 158degrees, not greater than about 156 degrees, not greater than about 154degrees, not greater than about 152 degrees, not greater than about 150degrees, not greater than about 148 degrees, not greater than about 146degrees, not greater than about 144 degrees, not greater than about 142degrees, or even not greater than about 140 degrees. It will beappreciated that the first interior angle 1247 can have a value within arange between any of the minimum and maximum values noted above.

The second interior corner 1246, along with the first curved section1242 and the second linear section 1243, can define the second interiorangle 1248, which can have an obtuse value. The second interior angle1248 can be measured as the angle formed between the second linearsection 1243 and a tangent 1284 of the first curved section 1242extending from the second interior corner 1246. According to oneembodiment, the second interior angle 1248 can have a value between atleast about 92 degrees and not greater than about 178 degrees. Moreparticularly, in at least one embodiment, the second interior angle 1248can have a value of at least about 94 degrees, such as at least about 96degrees, at least about 98 degrees, at least about 100 degrees, at leastabout 102 degrees, at least about 104 degrees, at least about 106degrees, at least about 108 degrees, at least about 110 degrees, atleast about 112 degrees, at least about 124 degrees, at least about 126degrees, at least about 128 degrees, at least about 120 degrees, atleast about 122 degrees, at least about 124 degrees, at least about 126degrees, at least about 128 degrees, at least about 130 degrees, atleast about 132 degrees, at least about 134 degrees, at least about 136degrees, at least about 138 degrees, or even at least about 140 degrees.In yet another embodiment, the second interior angle 1248 can have avalue of not greater than about 176 degrees, such as not greater thanabout 174 degrees, not greater than about 172 degrees, not greater thanabout 170 degrees, not greater than about 168 degrees, not greater thanabout 166 degrees, not greater than about 164 degrees, not greater thanabout 162 degrees, not greater than about 160 degrees, not greater thanabout 158 degrees, not greater than about 156 degrees, not greater thanabout 154 degrees, not greater than about 152 degrees, not greater thanabout 150 degrees, not greater than about 148 degrees, not greater thanabout 146 degrees, not greater than about 144 degrees, not greater thanabout 142 degrees, or even not greater than about 140 degrees. It willbe appreciated that the second interior angle 1248 can have a valuewithin a range between any of the minimum and maximum values notedabove.

As further illustrated, the first curved section 1242 of the firstportion 1206 of the side surface 1205 can have a substantially concaveshape and may curve inwards into the body 1201 toward the midpoint 1281.The first curved section 1242 may define an arc having a single distinctcurvature as illustrated in FIGS. 12A and 12B.

Moreover, the first curved section 1242 can have a particular radius ofcurvature (Rc1) relative to the width (W) (e.g., the total length (Lfp1)in an embodiment) of the body 1201 that may facilitate improvedperformance of the body. The radius of curvature may be determined bysuperimposing a best fit circle to the curvature of the first curvedsection 1242 and determining the radius of the best fit circle. Anysuitable computer program, such as ImageJ may be used in conjunctionwith an image (e.g., SEM image or light microscope image) of suitablemagnification of the body 1201 to accurately measure the best fitcircle. According to one embodiment, the first curved section 1242 canhave a radius of curvature (Rc1) that is at least half of the width (W)of the body 1201, such as at least about 0.8 times the width (W) of thebody 1201, at least 1.5 times the width (W) of the body 1201, or even atleast 2 times the width (W) of the body 1201. In another embodiment, theradius of curvature (Rc1) can be not greater than about 50 times thewidth (W) of the body 1201, such as not greater than about 20 times thewidth (W) of the body 1201, not greater than about 15 times the width(W) of the body 1201, not greater than about 10 times the width (W) ofthe body 1201, or even not greater than about 5 times the width (W) ofthe body 1201. The first curved section 1242 can have a radius ofcurvature (Rc1) within a range between any of the minimum and maximumvalues noted above.

In at least one embodiment, the first curved section 1242 can have aradius of curvature (Rc1) that is not greater than 4 mm or not greaterthan 3 mm or not greater than 2.5 mm or not greater than 2 mm or evennot greater than 1.5 mm. Still, in another embodiment, the first curvedsection 1242 can have a radius of curvature of at least 0.01 mm, such asat least 0.1 mm or at least 0.5 mm or at least 0.8 mm or even at least 1mm. It will be appreciated that the radius of curvature of any one ofthe curved sections described in the embodiments herein can be within arange including any of the minimum and maximum values noted above.

However, it will be appreciated that a particular side portion of a sidesurface can include multiple curved sections. For example, FIG. 13includes an illustration of one embodiment of a shaped abrasive particle1300 including a body 1301 with a first portion 1306 of a side surface1305. The first portion 1306 can include a first curved section 1342disposed between a first linear section 1341 and a second linear section1343. Moreover, the first portion 1306 can include a second curvedsection 1344 disposed between the second linear section 1343 and a thirdlinear section 1345, which second curved section 1344 may be spacedapart from the first curved section 1342. The linear sections 1341,1343, and 1345 can have any of the features of any linear sectionsdescribed in the embodiments herein. Similarly, the curved sections 1342and 1344 can have any of the features of the embodiments of the curvedsections described herein.

It will be further appreciated that it is within the scope ofembodiments herein to form a body of a shaped abrasive particle having acurved section with multiple curvatures, such that it has at least twodistinct curvatures. For example, FIG. 14 includes an illustration of afirst curved section 1442 having a first section 1443 defining a firstcurvature and a second section 1444 defining a second curvature.Notably, the curvature of the first section 1443 can be distinct fromthe curvature of the second section 1444. Moreover, the first section1443 can be integrally joined to the second section 1444.

FIG. 15 includes a top view of a shaped abrasive particle 1500 accordingto an embodiment. In particular, the shaped abrasive 1500 can include abody 1501 having the features of other shaped abrasive particles ofembodiments herein, including an upper major surface 1503 and a bottommajor surface (not shown) opposite the upper major surface 1503. Theupper major surface 1503 and the bottom major surface can be separatedfrom each other by at least one side surface 1505, which may include oneor more discrete side surface portions, including for example, a firstportion 1506 of the side surface 1505, a second portion 1507 of the sidesurface 1505, and a third portion 1508 of the side surface 1505. Inparticular, the first portion 1506 of the side surface 1505 can extendbetween a first corner 1509 and a second corner 1510. The second portion1507 of the side surface 1505 can extend between the second corner 1510and a third corner 1511. Notably, the second corner 1510 can be anexternal corner joining two portions of the side surface 1505. Thesecond corner 1510 and third corner 1511, which is also an externalcorner, are adjacent to each other and have no other external cornersdisposed between them. Also, the third portion 1508 of the side surface1505 can extend between the third corner 1511 and the first corner 1509,which are both external corners that are adjacent to each other havingno other external corners disposed between them.

As illustrated, the body 1501 can include a first portion 1506 includinga first curved section 1542 disposed between a first linear section 1541and a second linear section 1543 and between the external corners 1509and 1510. The body 1501 can further include a second portion 1507separated from the first portion 1506 of the side surface 1505 by theexternal corner 1510. The second portion 1507 of the side surface 1505can include a second curved section 1552 joining a third linear section1551 and a fourth linear section 1553.

The body 1501 can have any of the features of other embodiments herein,including but not limited to, a primary aspect ratio, a secondary aspectratio, a tertiary aspect ratio, and the like. In one aspect, the body1501 of the shaped abrasive particle 1500 can have a second portion 1507of the side surface 1505 with a partially-concave shape. A partiallyconcave shape can include the curved section 1552 that extends for afraction of the total length (Lfp2) of the second portion 1507 of theside surface 1505 between the adjacent corners 1510 and 1511. In anembodiment, the total length (Lfp2) can be equivalent to a width (W) ofthe body 1501. Moreover, as further illustrated in the embodiment ofFIG. 15, the second curved section 1552 can be disposed between thethird linear section 1551 and the fourth linear section 1553. The thirdlinear section 1551 can terminate at a first end at the first externalcorner 1510 of the body 1501, extend along the second portion 1507 ofthe side surface 1505 for a length (Ll3), and terminate at the joiningof the second portion 1507 with the second curved section 1552. Thesecond curved section 1552 and the third linear section 1551 can definea third interior corner 1554, which along with the second curved section1552 and the third linear section 1551 can define a first interior angle1555 having any of the features of the interior angles of embodimentsherein (e.g., defining an obtuse angle). The fourth linear section 1553can terminate at first end at the third external corner 1511 of the body1501, extend along the second portion 1507 of the side surface 1505 fora length (Ll4), and terminate at the joining of the second portion 1507with the second curved section 1552. The fourth linear section 1553 andthe second curved section 1552 can define a fourth interior corner 1556.The fourth interior corner 1556, along with the second curved section1552 and the fourth linear section 1553 can define a fourth interiorangle 1557 having any of the features of the interior angles ofembodiments herein (e.g., defining an obtuse angle).

As will be appreciated, the third linear section 1551 and the fourthlinear section 1553 can be substantially linear when viewed from the topdown as illustrated in FIG. 15. The second curved section 1552 can havea significant arcuate contour when viewed from the topdown, also asshown in FIG. 15. The third linear section 1551 can have a third linearsection length (Ll3) and can have any of the features of any of thelinear sections of shaped abrasive particles of the embodiments herein.The second curved section 1552 can have a length (Lc2) and can have anyof the features of the curved sections of shaped abrasive particles ofthe embodiments herein. The fourth linear section 1553 can have a length(Ll4) and can have any of the features of any of the linear sections ofshaped abrasive particles of the embodiments herein.

The body 1501 can include a first arm 1571 extending between themidpoint 1581 of the body 1501 and the terminal end of the first arm1571, defined by the external corner 1510. The first arm 1571 can have afirst arm axis 1572 extending between the terminal end of the first arm1571 and the midpoint 1581 and defining a total length (Larm1) of thefirst arm 1571.

According to one particular embodiment, the body 1501 can have a firstmaximum tip width (Wt1) that defines a maximum width of a first arm 1571of the body 1501 at a location between the terminal end (i.e., externalcorner 1510) of the first arm 1571 and the midpoint 1581. Notably, thefirst maximum tip width (Wt1) can be spaced a distance from the midpoint1581 along the first arm axis 1572 and spaced a distance from theterminal end of the first arm 1571 along the first arm axis 1572.Moreover, the first maximum tip width (Wt1) can define a first maximumtip width location 1573 along the first arm axis 1572.

The distance between the terminal end of the first arm 1571 and thefirst maximum tip width location 1573 can define a first tip length(Ltip1). The first tip length (Ltip1) can have a particular relationshiprelative to the length of the first arm 1571 (Larm1) that may facilitateimproved performance of the shaped abrasive particle 1500. In oneembodiment, the first tip length (Ltip1) can be a fraction of the totallength of the first arm (Larm1). For example, in one embodiment, thefirst tip length (Ltip1) can be at least about 0.01 (Larm1), such as atleast about 0.02 (Larm1), at least about 0.03 (Larm1), at least about0.04 (Larm1), at least about 0.05 (Larm1), at least about 0.06 (Larm1),at least about 0.07 (Larm1), at least about 0.08 (Larm1), at least about0.09 (Larm1), at least about 0.1 (Larm1), at least about 0.12 (Larm1),at least about 0.15 (Larm1), at least about 0.18 (Larm1), at least about0.2 (Larm1), at least about 0.22 (Larm1), at least about 0.25 (Larm1),at least about 0.28 (Larm1), at least about 0.3 (Larm1), at least about0.32 (Larm1), at least about 0.35 (Larm1), at least about 0.38 (Larm1),or even at least about 0.4 (Larm1). In another non-limiting embodiment,the first tip length (Ltip1) can be not greater than about 0.95 (Larm1),such as not greater than about 0.9 (Larm1), not greater than about 0.85(Larm1), not greater than about 0.8 (Larm1), not greater than about 0.75(Larm1), not greater than about 0.7 (Larm1), not greater than about 0.65(Larm1), not greater than about 0.6 (Larm1), not greater than about 0.55(Larm1), not greater than about 0.5 (Larm1), not greater than about 0.45(Larm1). It will be appreciated that the first tip length (Ltip1) can bewithin a range between any of the minimum and maximum values above.

The body 1501 can further include a first throat width (Wth1), which candefine a narrowest portion of the first arm between a first maximum tipwidth location 1573 of the first arm 1571 and the midpoint 1581.Moreover, the first throat width (Wth1) can define a first throat widthlocation 1574 along the axis 1572 of the first arm 1571. As illustrated,in certain embodiments, the first throat location 1574 can be closer tothe midpoint 1581 than the first maximum tip width location 1573.

In at least one embodiment, the distance between the first throat widthlocation 1574 and the first maximum tip width location 1573 can define afirst throat length (Lth1). In certain embodiments, the first throatlength (Lth1) can have a particular length relative to the length of thefirst arm (Larm1) that may improve the performance of the shapedabrasive particle 1500. For example, the first throat length (Lth1) canbe a fraction of a total length of the first arm (Larm1). In oneinstance, the first throat length (Lth1) can be at least about 0.01(Larm1), such as at least about 0.02 (Larm1), at least about 0.03(Larm1), at least about 0.04 (Larm1), at least about 0.05 (Larm1), atleast about 0.06 (Larm1), at least about 0.07 (Larm1), at least about0.08 (Larm1), at least about 0.09 (Larm1), at least about 0.1 (Larm1),at least about 0.12 (Larm1), at least about 0.15 (Larm1), at least about0.18 (Larm1), at least about 0.2 (Larm1), at least about 0.22 (Larm1),at least about 0.25 (Larm1), at least about 0.28 (Larm1), at least about0.3 (Larm1), at least about 0.32 (Larm1), at least about 0.35 (Larm1),at least about 0.38 (Larm1), at least about 0.4 (Larm1). In anothernon-limiting embodiment, the first throat length (Lth1) can be notgreater than about 0.95 (Larm1), such as not greater than about 0.9(Larm1), not greater than about 0.85 (Larm1), not greater than about 0.8(Larm1), not greater than about 0.75 (Larm1), not greater than about 0.7(Larm1), not greater than about 0.65 (Larm1), not greater than about 0.6(Larm1), not greater than about 0.55 (Larm1), not greater than about 0.5(Larm1), not greater than about 0.45 (Larm1). It will be appreciatedthat the first throat length (Lth1) can be within a range between any ofthe minimum and maximum values above.

The body 1501 can be formed such that the first throat width (Wth1) andthe maximum tip width (Wt1) have a particular relationship relative toeach other, which may improve the performance of the shaped abrasiveparticle 1500. For example, the first throat width (Wth1) can be lessthan the first maximum tip width (Wt1). In more particular instances,the first throat width (Wth1) can be not greater than about 0.95 (Wt1),such as not greater than about 0.9 (Wt1), not greater than about 0.85(Wt1), not greater than about 0.8 (Wt1), not greater than about 0.75(Wt1), not greater than about 0.7 (Wt1), not greater than about 0.65(Wt1), not greater than about 0.6 (Wt1), not greater than about 0.55(Wt1), not greater than about 0.5 (Wt1), or even not greater than about0.45 (Wt1). In yet another embodiment, the first throat width (Wth1) canbe at least about 0.01 (Wt1), such as at least about 0.05 (Wt1), atleast about 0.08 (Wt1), at least about 0.1 (Wt1), at least about 0.12(Wt1), at least about 0.15 (Wt1), at least about 0.18 (Wt1), at leastabout 0.2 (Wt1), at least about 0.22 (Wt1), at least about 0.25 (Wt1),at least about 0.28 (Wt1), at least about 0.3 (Wt1), at least about 0.32(Wt1), at least about 0.35 (Wt1), at least about 0.38 (Wt1), at leastabout 0.4 (Wt1), at least about 0.42 (Wt1), at least about 0.45 (Wt1),at least about 0.48 (Wt1), or even at least about 0.5 (Wt1). The firstthroat width (Wth1) can have a width relative to the first maximum tipwidth (Wt1) within a range between any of the minimum and maximum valuesnoted above.

FIG. 16 includes a top view illustration of a shaped abrasive particleaccording to an embodiment. In particular, the shaped abrasive particle1600 can include a body 1601 having the features of other shapedabrasive particles of embodiments herein, including an upper majorsurface 1603 and a bottom major surface (not shown) opposite the uppermajor surface 1603. The upper major surface 1603 and the bottom majorsurface can be separated from each other by at least one side surface1605, which may include one or more discrete side surface portions,including for example, a first portion 1606 of the side surface 1605, asecond portion 1607 of the side surface 1605, and a third portion 1608of the side surface 1605. In particular, the first portion 1606 of theside surface 1605 can extend between a first corner 1609 and a secondcorner 1610. The second portion 1607 of the side surface 1605 can extendbetween the second corner 1610 and a third corner 1611. Notably, thesecond corner 1610 can be an external corner joining two portions of theside surface 1605. The second corner 1610 and a third corner 1611, whichis also an external corner, are adjacent to each other and have no otherexternal corners disposed between them. Also, the third portion 1608 ofthe side surface 1605 can extend between the third corner 1611 and thefirst corner 1609, which are both external corners that are adjacent toeach other and have no other external corners disposed between them.

As illustrated, the body 1601 can include a first portion 1606 includinga first curved section 1642 disposed between a first linear section 1641and a second linear section 1643 and between the external corners 1609and 1610. The second portion 1607 is separated from the first portion1606 of the side surface 1605 by the external corner 1610. The secondportion 1607 of the side surface 1605 can include a second curvedsection 1652 joining a third linear section 1651 and a fourth linearsection 1653. Furthermore, the body 1601 can include a third portion1608 separated from the first portion 1606 of the side surface 1605 bythe external corner 1609 and separated from the second portion 1607 bythe external corner 1611. The third portion 1608 of the side surface1605 can include a third curved section 1662 joining a fifth linearsection 1661 and a sixth linear section 1663.

The body 1601 can have any of the features of other embodiments herein,including but not limited to, a primary aspect ratio, a secondary aspectratio, a tertiary aspect ratio, and the like. In one aspect, the body1601 of the shaped abrasive particle 1600 can have a third portion 1608of the side surface 1605 with a partially-concave shape. A partiallyconcave shape can include the third curved section 1662 that extends fora fraction of the total length (Lfp3) of the third portion 1608 of theside surface 1605 between the adjacent, external corners 1609 and 1611.In an embodiment, the total length (Lfp3) can be equivalent to a width(W) of the body 1601. Moreover, as further illustrated in the embodimentof FIG. 16, the third curved section 1662 can be disposed between thefifth linear section 1661 and the sixth linear section 1663. The fifthlinear section 1661 can terminate at a first end at the first externalcorner 1611 of the body 1601, extend along the third portion 1608 of theside surface 1605 for a length (Ll5), and terminate at a second end atthe joining of the third portion 1608 with the second curved section1662. The third curved section 1662 and the fifth linear section 1661can define a sixth interior corner 1664, which along with the thirdcurved section 1662 and the fifth linear section 1661 can define a fifthinterior angle 1665 having any of the features of the interior angles ofembodiments herein (e.g., defining an obtuse angle). The sixth linearsection 1663 can terminate at a first end at the external corner 1609 ofthe body 1601, extend along the third portion 1608 of the side surface1605 for a length (Ll6), and terminate at a second end at the joining ofthe third portion 1608 with the third curved section 1662. The sixthlinear section 1663 and the third curved section 1662 can define a sixthinterior corner 1666. The sixth interior corner 1666, along with thethird curved section 1662 and the sixth linear section 1663, can definea sixth interior angle 1667 having any of the features of the interiorangles of embodiments herein (e.g., defining an obtuse angle).

As will be appreciated, the fifth linear section 1661 and the sixthlinear section 1663 can be substantially linear when viewed from the topdown as illustrated in FIG. 16. The third curved section 1662 can have asignificant arcuate contour, also when viewed from the top down as shownin FIG. 16. The fifth linear section 1661 can have a fifth linearsection length (Ll5) and can have any of the features of any of thelinear sections of shaped abrasive particles of the embodiments herein.The third curved section 1662 can have a length (Lc3) and can have anyof the features of the curved sections of shaped abrasive particles ofthe embodiments herein. The sixth linear section 1663 can have a length(Ll6) and can have any of the features of any of the linear sections ofshaped abrasive particles of the embodiments herein.

The body 1605 can include a first arm 1671 extending between themidpoint 1681 of the body 1601 and the terminal end (e.g., the externalcorner 1610) of the first arm 1671. The first arm 1671 can have any ofthe features of the arms of the embodiments herein, including forexample, but not limited to, a maximum tip width, a throat width, afirst tip length, a first throat length, and the like. The body 1605 caninclude a second arm 1692, extending between the midpoint 1681 of thebody 1601 and the terminal end (e.g., the external corner 1611) of thesecond arm 1692. The second arm 1692 can have any of the features of thearms of the embodiments herein, including for example, but not limitedto, a maximum tip width, a throat width, a first tip length, a firstthroat length, and the like. As also illustrated in FIG. 16, the body1601 can include a third arm 1693, extending between the midpoint 1681of the body 1601 and the terminal end (e.g., the external corner 1609)of the third arm 1693. The third arm 1693 can have any of the featuresof the arms of the embodiments herein, including for example, but notlimited to, a maximum tip width, a throat width, a first tip length, afirst throat length, and the like.

In one aspect, the body 1601 can have a first maximum tip width (Wt1)that defines a maximum width of the first arm 1671 of the body 1601 at alocation between the terminal end (e.g., external corner 1610) of thefirst arm 1671 and the midpoint 1681. Notably, the first maximum tipwidth (Wt1) can be spaced a distance from the midpoint 1681 along afirst arm axis 1672 and spaced a distance from the terminal end of thefirst arm 1671 along the first arm axis 1672. Moreover, the firstmaximum tip width (Wt1) can define a first maximum tip width location1673 along the first arm axis 1672. The distance between the terminalend of the first arm 1671 and the first maximum tip width location 1673can define a first tip length (Ltip1). The first tip length (Ltip1) canhave a particular relationship relative to a length of the first arm1671 (referred to as Larm1 defined as a total length between theterminal end of the first arm and the midpoint 1681, which mayfacilitate improved performance of the shaped abrasive particle. Thefirst arm 1671 can have a first arm axis 1672 extending between theterminal end of the first arm and the midpoint 1681 and defining a totallength (Larm1) of the first arm 1671. In one embodiment, the first tiplength (Ltip1) can be a fraction of a total length of the first arm(Larm1). For example, in one embodiment, the first tip length (Ltip1)can be at least about 0.01 (Larm1), such as at least about 0.02 (Larm1),at least about 0.03 (Larm1), at least about 0.04 (Larm1), at least about0.05 (Larm1), at least about 0.06 (Larm1), at least about 0.07 (Larm1),at least about 0.08 (Larm1), at least about 0.09 (Larm1), at least about0.1 (Larm1), at least about 0.12 (Larm1), at least about 0.15 (Larm1),at least about 0.18 (Larm1), at least about 0.2 (Larm1), at least about0.22 (Larm1), at least about 0.25 (Larm1), at least about 0.28 (Larm1),at least about 0.3 (Larm1), at least about 0.32 (Larm1), at least about0.35 (Larm1), at least about 0.38 (Larm1), or even at least about 0.4(Larm1). In another non-limiting embodiment, the first tip length(Ltip1) can be not greater than about 0.95 (Larm1), such as not greaterthan about 0.9 (Larm1), not greater than about 0.85 (Larm1), not greaterthan about 0.8 (Larm1), not greater than about 0.75 (Larm1), not greaterthan about 0.7 (Larm1), not greater than about 0.65 (Larm1), not greaterthan about 0.6 (Larm1), not greater than about 0.55 (Larm1), not greaterthan about 0.5 (Larm1), not greater than about 0.45 (Larm1). It will beappreciated that the first tip length (Ltip1) can be within a rangebetween any of the minimum and maximum values above.

The body 1601 can further include a first throat width (Wth1), which candefine a narrowest portion of the first arm 1671 between a first maximumtip width location 1673 of the first arm 1671 and the midpoint 1681.Moreover, the first throat width (Wth1) can define a first throat widthlocation 1674 along the axis 1672 of the first arm 1671. As illustrated,in certain embodiments, the first throat location 1674 can be closer tothe midpoint 1681 than the first maximum tip width location 1673.

In at least one embodiment, the distance between the first throatlocation 1674 and the first maximum tip width location 1673 can define afirst throat length (Lth1). In certain embodiments, the first throatlength (Lth1) can have a particular length relative to the length of thefirst arm (Larm1) that may improve the performance of the shapedabrasive particle 1600. For example, the first throat length (Lth1) canbe a fraction of a total length of the first arm (Larm1). In oneinstance, the first throat length (Lth1) can be at least about 0.01(Larm1), such as at least about 0.02 (Larm1), at least about 0.03(Larm1), at least about 0.04 (Larm1), at least about 0.05 (Larm1), atleast about 0.06 (Larm1), at least about 0.07 (Larm1), at least about0.08 (Larm1), at least about 0.09 (Larm1), at least about 0.1 (Larm1),at least about 0.12 (Larm1), at least about 0.15 (Larm1), at least about0.18 (Larm1), at least about 0.2 (Larm1), at least about 0.22 (Larm1),at least about 0.25 (Larm1), at least about 0.28 (Larm1), at least about0.3 (Larm1), at least about 0.32 (Larm1), at least about 0.35 (Larm1),at least about 0.38 (Larm1), at least about 0.4 (Larm1). In anothernon-limiting embodiment, the first throat length (Lth1) can be notgreater than about 0.95 (Larm1), such as not greater than about 0.9(Larm1), not greater than about 0.85 (Larm1), not greater than about 0.8(Larm1), not greater than about 0.75 (Larm1), not greater than about 0.7(Larm1), not greater than about 0.65 (Larm1), not greater than about 0.6(Larm1), not greater than about 0.55 (Larm1), not greater than about 0.5(Larm1), not greater than about 0.45 (Larm1). It will be appreciatedthat the first throat length (Lth1) can be within a range between any ofthe minimum and maximum values above.

The body 1601 can be formed such that the first throat width (Wth1) andthe first maximum tip width (Wt1) have a particular relationshiprelative to each other, which may improve the performance of the shapedabrasive particle 1600. For example, the first throat width (Wth1) canbe less than the first maximum tip width (Wt1). In more particularinstances, the first throat width (Wth1) can be not greater than about0.95 (Wt1), such as not greater than about 0.9 (Wt1), not greater thanabout 0.85 (Wt1), not greater than about 0.8 (Wt1), not greater thanabout 0.75 (Wt1), not greater than about 0.7 (Wt1), not greater thanabout 0.65 (Wt1), not greater than about 0.6 (Wt1), not greater thanabout 0.55 (Wt1), not greater than about 0.5 (Wt1), or even not greaterthan about 0.45 (Wt1). In yet another embodiment, the first throat width(Wth1) can be at least about 0.01 (Wt1), such as at least about 0.05(Wt1), at least about 0.08 (Wt1), at least about 0.1 (Wt1), at leastabout 0.12 (Wt1), at least about 0.15 (Wt1), at least about 0.18 (Wt1),at least about 0.2 (Wt1), at least about 0.22 (Wt1), at least about 0.25(Wt1), at least about 0.28 (Wt1), at least about 0.3 (Wt1), at leastabout 0.32 (Wt1), at least about 0.35 (Wt1), at least about 0.38 (Wt1),at least about 0.4 (Wt1), at least about 0.42 (Wt1), at least about 0.45(Wt1), at least about 0.48 (Wt1), or even at least about 0.5 (Wt1). Thefirst throat width (Wth1) can have a width relative to the first maximumtip width (Wt1) within a range between any of the minimum and maximumvalues noted above.

In one aspect, the body 1601 can have a second maximum tip width (Wt2)that defines a maximum width of the second arm 1692 of the body 1601 ata location between the terminal end (e.g., external corner 1611) of thesecond arm 1692 and the midpoint 1681. Notably, the second maximum tipwidth (Wt2) can be spaced a distance from the midpoint 1681 along thesecond arm axis 1682 and spaced a distance from the terminal end of thesecond arm 1692 along the second arm axis 1682. Moreover, the secondmaximum tip width (Wt2) can define a second maximum tip width location1675 along the second arm axis 1682. The distance between the terminalend of the second arm 1692 and the second maximum tip width location1675 can define a second tip length (Ltip2). The second tip length(Ltip2) can have a particular relationship relative to a length of thesecond arm 1692 (referred to generally as Larm2), defined as a totallength between the terminal end of the second arm 1692 and the midpoint1681, which may facilitate improved performance of the shaped abrasiveparticle 1600. In one embodiment, the second tip length (Ltip2) can be afraction of a total length of the second arm (Larm2). For example, inone embodiment, the second tip length (Ltip2) can be at least about 0.01(Larm2), such as at least about 0.02 (Larm2), at least about 0.03(Larm2), at least about 0.04 (Larm2), at least about 0.05 (Larm2), atleast about 0.06 (Larm2), at least about 0.07 (Larm2), at least about0.08 (Larm2), at least about 0.09 (Larm2), at least about 0.1 (Larm2),at least about 0.12 (Larm2), at least about 0.15 (Larm2), at least about0.18 (Larm2), at least about 0.2 (Larm2), at least about 0.22 (Larm2),at least about 0.25 (Larm2), at least about 0.28 (Larm2), at least about0.3 (Larm2), at least about 0.32 (Larm2), at least about 0.35 (Larm2),at least about 0.38 (Larm2), or even at least about 0.4 (Larm2). Inanother non-limiting embodiment, the second tip length (Ltip2) can benot greater than about 0.95 (Larm2), such as not greater than about 0.9(Larm2), not greater than about 0.85 (Larm2), not greater than about 0.8(Larm2), not greater than about 0.75 (Larm2), not greater than about 0.7(Larm2), not greater than about 0.65 (Larm2), not greater than about 0.6(Larm2), not greater than about 0.55 (Larm2), not greater than about 0.5(Larm2), not greater than about 0.45 (Larm2). It will be appreciatedthat the second tip length (Ltip2) can be within a range between any ofthe minimum and maximum values above.

The body 1601 can further include a second throat width (Wth2), whichcan define a narrowest portion of the second arm 1692 between a secondmaximum tip width location 1675 of the second arm 1692 and the midpoint1681. Moreover, the second throat width (Wth2) can define a secondthroat width location 1676 along the axis 1682 of the second arm 1692.As illustrated, in certain embodiments, the second throat width location1676 can be closer to the midpoint 1681 than the second maximum tipwidth location 1675.

In at least one embodiment, the distance between the second throatlocation 1676 and the second maximum tip width location 1675 can definea second throat length (Lth2). In certain embodiments, the second throatlength (Lth2) can have a particular length relative to the length of thesecond arm (Larm2) that may improve the performance of the shapedabrasive particle 1600. For example, the second throat length (Lth2) canbe a fraction of a total length of the second arm (Larm2). In oneinstance, the second throat length (Lth2) can be at least about 0.01(Larm2), such as at least about 0.02 (Larm2), at least about 0.03(Larm2), at least about 0.04 (Larm2), at least about 0.05 (Larm2), atleast about 0.06 (Larm2), at least about 0.07 (Larm2), at least about0.08 (Larm2), at least about 0.09 (Larm2), at least about 0.1 (Larm2),at least about 0.12 (Larm2), at least about 0.15 (Larm2), at least about0.18 (Larm2), at least about 0.2 (Larm2), at least about 0.22 (Larm2),at least about 0.25 (Larm2), at least about 0.28 (Larm2), at least about0.3 (Larm2), at least about 0.32 (Larm2), at least about 0.35 (Larm2),at least about 0.38 (Larm2), at least about 0.4 (Larm2). In anothernon-limiting embodiment, the second throat length (Lth2) can be notgreater than about 0.95 (Larm2), such as not greater than about 0.9(Larm2), not greater than about 0.85 (Larm2), not greater than about 0.8(Larm2), not greater than about 0.75 (Larm2), not greater than about 0.7(Larm2), not greater than about 0.65 (Larm2), not greater than about 0.6(Larm2), not greater than about 0.55 (Larm2), not greater than about 0.5(Larm2), not greater than about 0.45 (Larm2). It will be appreciatedthat the second throat length (Lth2) can be within a range between anyof the minimum and maximum values above.

The body 1601 can be formed such that the second throat width (Wth2) andthe second maximum tip width (Wt2) have a particular relationshiprelative to each other, which may improve the performance of the shapedabrasive particle 1600. For example, the second throat width (Wth2) canbe less than the second maximum tip width (Wt2). In more particularinstances, the second throat width (Wth2) can be not greater than about0.95 (Wt2), such as not greater than about 0.9 (Wt2), not greater thanabout 0.85 (Wt2), not greater than about 0.8 (Wt2), not greater thanabout 0.75 (Wt2), not greater than about 0.7 (Wt2), not greater thanabout 0.65 (Wt2), not greater than about 0.6 (Wt2), not greater thanabout 0.55 (Wt2), not greater than about 0.5 (Wt2), or even not greaterthan about 0.45 (Wt2). In yet another embodiment, the second throatwidth (Wth2) can be at least about 0.01 (Wt2), such as at least about0.05 (Wt2), at least about 0.08 (Wt2), at least about 0.1 (Wt2), atleast about 0.12 (Wt2), at least about 0.15 (Wt2), at least about 0.18(Wt2), at least about 0.2 (Wt2), at least about 0.22 (Wt2), at leastabout 0.25 (Wt2), at least about 0.28 (Wt2), at least about 0.3 (Wt2),at least about 0.32 (Wt2), at least about 0.35 (Wt2), at least about0.38 (Wt2), at least about 0.4 (Wt2), at least about 0.42 (Wt2), atleast about 0.45 (Wt2), at least about 0.48 (Wt2), or even at leastabout 0.5 (Wt2). The second throat width (Wth2) can have a widthrelative to the second maximum tip width (Wt2) within a range betweenany of the minimum and maximum values noted above.

In yet another aspect, the body 1601 can have a third maximum tip width(Wt3) that defines a maximum width of the third arm 1693 of the body1601 at a location between the terminal end (e.g., external corner 1609)of the third arm 1693 and the midpoint 1681. Notably, the third maximumtip width (Wt3) can be spaced a distance from the midpoint 1681 alongthe third arm axis 1683 and spaced a distance from the terminal end ofthe third arm 1693 along the third arm axis 1683. Moreover, the thirdmaximum tip width (Wt3) can define a third maximum tip width location1677 along the third arm axis 1683. The distance between the terminalend of the third arm 1693 and the third maximum tip width location 1677can define a third tip length (Ltip3). The third tip length (Ltip3) canhave a particular relationship relative to the length of the third arm1693 (referred to generally as Larm3), defined as a total length betweenthe terminal end of the third arm 1693 and the midpoint 1681, which mayfacilitate improved performance of the shaped abrasive particle 1600. Inone embodiment, the third tip length (Ltip3) can be a fraction of atotal length of the third arm (Larm3). For example, in one embodiment,the third tip length (Ltip3) can be at least about 0.01 (Larm3), such asat least about 0.02 (Larm3), at least about 0.03 (Larm3), at least about0.04 (Larm3), at least about 0.05 (Larm3), at least about 0.06 (Larm3),at least about 0.07 (Larm3), at least about 0.08 (Larm3), at least about0.09 (Larm3), at least about 0.1 (Larm3), at least about 0.12 (Larm3),at least about 0.15 (Larm3), at least about 0.18 (Larm3), at least about0.2 (Larm3), at least about 0.22 (Larm3), at least about 0.25 (Larm3),at least about 0.28 (Larm3), at least about 0.3 (Larm3), at least about0.32 (Larm3), at least about 0.35 (Larm3), at least about 0.38 (Larm3),or even at least about 0.4 (Larm3). In another non-limiting embodiment,the third tip length (Ltip3) can be not greater than about 0.95 (Larm3),such as not greater than about 0.9 (Larm3), not greater than about 0.85(Larm3), not greater than about 0.8 (Larm3), not greater than about 0.75(Larm3), not greater than about 0.7 (Larm3), not greater than about 0.65(Larm3), not greater than about 0.6 (Larm3), not greater than about 0.55(Larm3), not greater than about 0.5 (Larm3), not greater than about 0.45(Larm3). It will be appreciated that the third tip length (Ltip3) can bewithin a range between any of the minimum and maximum values above.

The body 1601 can further include a third throat width (Wth3), which candefine a narrowest portion of the third arm 1693 between the thirdmaximum tip width location 1677 of the third arm 1693 and the midpoint1681. Moreover, the third throat width (Wth3) can define a third throatwidth location 1678 along the axis 1683 of the third arm 1693. Asillustrated, in certain embodiments, the third throat location 1678 canbe closer to the midpoint 1681 than the third maximum tip width location1677.

In at least one embodiment, the distance between the third throatlocation 1678 and the third maximum tip width location 1677 can define athird throat length (Lth3). In certain embodiments, the third throatlength (Lth3) can have a particular length relative to the length of thethird arm (Larm3) that may improve the performance of the shapedabrasive particle 1600. For example, the third throat length (Lth3) canbe a fraction of a total length of the third arm (Larm3). In oneinstance, the third throat length (Lth3) can be at least about 0.01(Larm3), such as at least about 0.02 (Larm3), at least about 0.03(Larm3), at least about 0.04 (Larm3), at least about 0.05 (Larm3), atleast about 0.06 (Larm3), at least about 0.07 (Larm3), at least about0.08 (Larm3), at least about 0.09 (Larm3), at least about 0.1 (Larm3),at least about 0.12 (Larm3), at least about 0.15 (Larm3), at least about0.18 (Larm3), at least about 0.2 (Larm3), at least about 0.22 (Larm3),at least about 0.25 (Larm3), at least about 0.28 (Larm3), at least about0.3 (Larm3), at least about 0.32 (Larm3), at least about 0.35 (Larm3),at least about 0.38 (Larm3), at least about 0.4 (Larm3). In anothernon-limiting embodiment, the third throat length (Lth3) can be notgreater than about 0.95 (Larm3), such as not greater than about 0.9(Larm3), not greater than about 0.85 (Larm3), not greater than about 0.8(Larm3), not greater than about 0.75 (Larm3), not greater than about 0.7(Larm3), not greater than about 0.65 (Larm3), not greater than about 0.6(Larm3), not greater than about 0.55 (Larm3), not greater than about 0.5(Larm3), not greater than about 0.45 (Larm3). It will be appreciatedthat the second throat length (Lth3) can be within a range between anyof the minimum and maximum values above.

The body 1601 can be formed such that the third throat width (Wth3) andthe third maximum tip width (Wt3) have a particular relationshiprelative to each other, which may improve the performance of the shapedabrasive particle 1600. For example, the third throat width (Wth3) canbe less than the third maximum tip width (Wt3). In more particularinstances, the third throat width (Wth3) can be not greater than about0.95 (Wt3), such as not greater than about 0.9 (Wt3), not greater thanabout 0.85 (Wt3), not greater than about 0.8 (Wt3), not greater thanabout 0.75 (Wt3), not greater than about 0.7 (Wt3), not greater thanabout 0.65 (Wt3), not greater than about 0.6 (Wt3), not greater thanabout 0.55 (Wt3), not greater than about 0.5 (Wt3), or even not greaterthan about 0.45 (Wt3). In yet another embodiment, the third throat width(Wth3) can be at least about 0.01 (Wt3), such as at least about 0.05(Wt3), at least about 0.08 (Wt3), at least about 0.1 (Wt3), at leastabout 0.12 (Wt3), at least about 0.15 (Wt3), at least about 0.18 (Wt3),at least about 0.2 (Wt3), at least about 0.22 (Wt3), at least about 0.25(Wt3), at least about 0.28 (Wt3), at least about 0.3 (Wt3), at leastabout 0.32 (Wt3), at least about 0.35 (Wt3), at least about 0.38 (Wt3),at least about 0.4 (Wt3), at least about 0.42 (Wt3), at least about 0.45(Wt3), at least about 0.48 (Wt3), or even at least about 0.5 (Wt3). Thethird throat width (Wth3) can have a width relative to the third maximumtip width (Wt3) within a range between any of the minimum and maximumvalues noted above.

FIG. 17 includes a top view illustration of a shaped abrasive particleaccording to alternative embodiment. In particular, the shaped abrasiveparticle 1700 can include a body 1701 having the features of othershaped abrasive particles of embodiments herein, including an uppermajor surface 1703 and a bottom major surface (not shown) opposite theupper major surface 1703. The upper major surface 1703 and the bottommajor surface can be separated from each other by at least one sidesurface 1705, which may include one or more discrete side surfaceportions, including for example, a first portion 1706 of the sidesurface 1705, a second portion 1707 of the side surface 1705, and athird portion 1708 of the side surface 1705. In particular, the firstportion 1706 of the side surface 1705 can extend between a first corner1709 and a second corner 1710. The second portion 1707 of the sidesurface 1705 can extend between the second corner 1710 and a thirdcorner 1711. Notably, the second corner 1710 can be an external cornerjoining two portions of the side surface 1705. The second corner 1710and a third corner 1711, which is also an external corner, are adjacentto each other and have no other external corners disposed between them.Also, the third portion 1708 of the side surface 1705 can extend betweenthe third corner 1711 and the first corner 1709, which are both externalcorners that are adjacent to each other and have no other externalcorners disposed between them.

As illustrated, the body 1701 can include a first portion 1706 includinga first curved section 1742 disposed between a first linear section 1741and a second linear section 1743 and between the external corners 1609and 1610. The second portion 1707 of the body 1701 can further include asecond curved section 1752 disposed between a third linear section 1751and a fourth linear section 1753 and between the external corners 1610and 1611. Moreover, the third portion 1708 can include a third curvedsection 1762 disposed between a fifth linear section 1761 and a sixthlinear section 1763 and between the external corners 1609 and 1611.Notably, the second curved section 1752 and the third curved section1762 have a different contour as compared to the first curved section1742. Accordingly, the first arm 1771 of the body 1701, which extendsbetween the midpoint 1781 of the body 1701 and the terminal end (i.e.,the external corner 1610) of the first arm 1771 can have a first maximumtip width (Wt1), a first maximum tip width location 1773, a first tiplength (Ltip1), a first throat width (Wth1), a first throat location1774, and a first throat length (Lth1) having any of the same featuresof the embodiments herein. Notably, the first arm 1771 can have a firstthroat width (Wth1) that is less than or equal to the first maximum tipwidth (Wt1). By contrast, given the curvatures of the second curvedsection 1752 and third curved section 1762, the second arm 1792, whichextends between the midpoint 1781 of the body 1701 and the terminal end(i.e., the external corner 1711) of the second arm 1792 does not have athroat width region, which is a region having a width that can be lessthan or equal to the second maximum tip width (Wt2) disposed between thesecond maximum tip width location 1766 and the midpoint 1781. However,as illustrated, the second arm 1792 can still have a maximum tip width(Wt2) extending between the interior corners 1784 and 1785, whichfurther defines the second tip length (Ltip2), which may have any of thefeatures of the embodiments herein. Moreover, as will be appreciated,the interior corner 1784 can define an angle 1782 that has a value thatis distinct from the angle 1780 defined by the interior corner 1779.

The first arm 1771 can have any of the features of the arms of theembodiments herein, including for example, but not limited to, a maximumtip width, a throat width, a first tip length, a first throat length,and the like. The body 1705 can include a second arm 1792, extendingbetween the midpoint 1781 of the body 1701 and the terminal end (e.g.,the external corner 1711) of the second arm 1792. The second arm 1792can have any of the features of the arms of the embodiments herein,including for example, but not limited to, a maximum tip width, a throatwidth, a first tip length, a first throat length, and the like. As alsoillustrated in FIG. 17, the body 1701 can include a third arm 1793,extending between the midpoint 1781 of the body 1701 and the terminalend (e.g., the external corner 1709) of the third arm 1793. The thirdarm 1793 can have any of the features of the arms of the embodimentsherein, including for example, but not limited to, a maximum tip width,a throat width, a first tip length, a first throat length, and the like.

The shaped abrasive particles of the embodiments herein may be formedusing any of the processes described herein. FIG. 18A includes a topview of a shaped abrasive particle 1800 according to an embodiment.Notably, the body 1801 may be formed such that it has a particularinterrelationship of at least three grain features, including apredetermined strength, a predetermined tip sharpness, and apredetermined Shape Index. It will be appreciated that while referenceis made to FIG. 18A, the grain features apply to all shaped abrasiveparticles of the embodiments herein. The tip sharpness of a shapedabrasive particle, which may be an average tip sharpness, may bemeasured by determining the radius of a best fit circle on an externalcorner of the body 1801. For example, turning to FIG. 18A, a top view ofthe upper major surface 1803 of the body 1801 is provided. At anexternal corner 1831, a best fit circle is overlaid on the image of thebody 1801 of the shaped abrasive particle 1800, and the radius of thebest fit circle relative to the curvature of the external corner 1831defines the value of tip sharpness for the external corner 1831. Themeasurement may be recreated for each external corner of the body 1801to determine the average individual tip sharpness for a single shapedabrasive particle 1800. Moreover, the measurement may be recreated on asuitable sample size of shaped abrasive particles of a batch of shapedabrasive particles to derive the average batch tip sharpness. Anysuitable computer program, such as ImageJ may be used in conjunctionwith an image (e.g., SEM image or light microscope image) of suitablemagnification to accurately measure the best fit circle and the tipsharpness.

The shaped abrasive particles of the embodiments herein may have aparticular tip sharpness that facilitates formation of shaped abrasiveparticles with a particular sharpness, strength and Shape Index factor(i.e., 3SF). For example, the body of a shaped abrasive particle,according to an embodiment, can have a tip sharpness within a rangebetween not greater than about 80 microns and at least about 1 micron.Moreover, in certain instances, the body can have a tip sharpness of notgreater than about 78 microns, such as not greater than about 76microns, not greater than about 74 microns, not greater than about 72microns, not greater than about 70 microns, not greater than about 68microns, not greater than about 66 microns, not greater than about 64microns, not greater than about 62 microns, not greater than about 60microns, not greater than about 58 microns, not greater than about 56microns, not greater than about 54 microns, not greater than about 52microns, not greater than about 50 microns, not greater than about 48microns, not greater than about 46 microns, not greater than about 44microns, not greater than about 42 microns, not greater than about 40microns, not greater than about 38 microns, not greater than about 36microns, not greater than about 34 microns, not greater than about 32microns, not greater than about 30 microns, not greater than about 38microns, not greater than about 36 microns, not greater than about 34microns, not greater than about 32 microns, not greater than about 30microns, not greater than about 28 microns, not greater than about 26microns, not greater than about 24 microns, not greater than about 22microns, not greater than about 20 microns, not greater than about 18microns, not greater than about 16 microns, not greater than about 14microns, not greater than about 12 microns, not greater than about 10microns. In yet another non-limiting embodiment, the tip sharpness canbe at least about 2 microns, such as at least about 4 microns, at leastabout 6 microns, at least about 8 microns, at least about 10 microns, atleast about 12 microns, at least about 14 microns, at least about 16microns, at least about 18 microns, at least about 20 microns, at leastabout 22 microns, at least about 24 microns, at least about 26 microns,at least about 28 microns, at least about 30 microns, at least about 32microns, at least about 34 microns, at least about 36 microns, at leastabout 38 microns, at least about 40 microns, at least about 42 microns,at least about 44 microns, at least about 46 microns, at least about 48microns, at least about 50 microns, at least about 52 microns, at leastabout 54 microns, at least about 56 microns, at least about 58 microns,at least about 60 microns, at least about 62 microns, at least about 64microns, at least about 66 microns, at least about 68 microns, at leastabout 70 microns. It will be appreciated that the body can have a tipsharpness within a range between any of the minimum and maximum valuesnoted above.

As noted above, another grain feature is the Shape Index. The ShapeIndex of the body 1801 can be described as a value of an outer radius ofa best-fit outer circle superimposed on the body, as viewed in twodimensions of a plane of length and width of the body 1801 (e.g., theupper major surface or the bottom major surface), compared to an innerradius of the largest best-fit inner circle that fits entirely withinthe body 1801, as viewed in the same plane of length and width. Forexample, turning to FIG. 18B, the shaped abrasive particle 1800 isprovided with two circles superimposed on the illustration todemonstrate the calculation of Shape Index. A first circle issuperimposed on the body 1801, which is a best-fit outer circlerepresenting the smallest circle that can be used to fit the entireperimeter of the body 1801 within its boundaries. The outer circle has aradius (Ro). For shapes such as that illustrated in FIG. 18B, the outercircle may intersect the perimeter of the body at each of the threeexternal corners. However, it will be appreciated that for certainirregular or complex shapes, the body may not fit uniformly within thecircle such that each of the corners intersect the circle at equalintervals, but a best-fit, outer circle still may be formed. Anysuitable computer program, such as ImageJ may be used in conjunctionwith an image of suitable magnification (e.g., SEM image or lightmicroscope image) to create the outer circle and measure the radius(Ro).

A second, inner circle can be superimposed on the body 1801, asillustrated in FIG. 18B, which circle is a best fit circle representingthe largest circle that can be placed entirely within the perimeter ofthe body 1801 as viewed in the plane of the length and width of the body1801. The inner circle can have a radius (Ri). It will be appreciatedthat for certain irregular or complex shapes, the inner circle may notfit uniformly within the body such that the perimeter of the circlecontacts portions of the body at equal intervals, such as shown for theshape of FIG. 18B. However, a best-fit, inner circle still may beformed. Any suitable computer program, such as ImageJ may be used inconjunction with an image of suitable magnification (e.g., SEM image orlight microscope image) to create the inner circle and measure theradius (Ri).

The Shape Index can be calculated by dividing the outer radius by theinner radius (i.e., Shape Index=Ri/Ro). For example, the body 1801 ofthe shaped abrasive particle 1800 has a Shape Index of approximately0.35.

The shaped abrasive particles of the embodiments herein may have aparticular Shape Index that facilitates formation of shaped abrasiveparticles with a particular 3SF. For example, the body 1801 may have aShape Index within a range between at least about 0.01 and not greaterthan about 0.49. More particularly, in one non-limiting embodiment, thebody 1801 of the shaped abrasive particle can have a Shape Index of atleast about 0.02, such as at least about 0.03, at least about 0.04, atleast about 0.05, at least about 0.06, at least about 0.07, at leastabout 0.08, at least about 0.09, at least about 0.10, at least about0.11, at least about 0.12, at least about 0.13, at least about 0.14, atleast about 0.15, at least about 0.16, at least about 0.17, at leastabout 0.18, at least about 0.19, at least about 0.20, at least about0.21, at least about 0.22, at least about 0.23, at least about 0.24, atleast about 0.25, at least about 0.26, at least about 0.27, at leastabout 0.28, at least about 0.29, at least about 0.30, at least about0.31, at least about 0.32, at least about 0.33, at least about 0.34, atleast about 0.35, at least about 0.36, at least about 0.37, at leastabout 0.38, at least about 0.39, at least about 0.40, at least about0.41, at least about 0.42, at least about 0.43, at least about 0.44, atleast about 0.45, at least about 0.46, or even at least about 0.47. Instill another non-limiting embodiment, the body 1801 can have a ShapeIndex of not greater than about 0.48, such as not greater than about0.47, not greater than about 0.46, not greater than about 0.45, notgreater than about 0.44, not greater than about 0.43, not greater thanabout 0.42, not greater than about 0.41, not greater than about 0.40,not greater than about 0.39, not greater than about 0.38, not greaterthan about 0.37, not greater than about 0.36, not greater than about0.35, not greater than about 0.34, not greater than about 0.33, notgreater than about 0.32, not greater than about 0.31, not greater thanabout 0.30, not greater than about 0.29, not greater than about 0.28,not greater than about 0.27, not greater than about 0.26, not greaterthan about 0.25, not greater than about 0.24, not greater than about0.23, not greater than about 0.22, not greater than about 0.21, notgreater than about 0.20, not greater than about 0.19, not greater thanabout 0.18, not greater than about 0.17, not greater than about 0.16,not greater than about 0.15, not greater than about 0.14, not greaterthan about 0.13, not greater than about 0.12, not greater than about0.11, not greater than about 0.10, not greater than about 0.09, notgreater than about 0.08, not greater than about 0.07, not greater thanabout 0.06, not greater than about 0.05, or even not greater than about0.04. It will be appreciated that the body 1801 can have a Shape Indexwithin a range between any of the minimum and maximum values notedabove.

Moreover, as noted herein, the body 1801 may be formed to have aparticular strength. The strength of the body may be measured viaHertzian indentation. In this method the abrasive grains are glued on aslotted aluminum SEM sample mounting stub. The slots are approximately250 μm deep and wide enough to accommodate the grains in a row. Thegrains are polished in an automatic polisher using a series of diamondpastes, with the finest paste of 1 μm to achieve a final mirror finish.At the final step, the polished grains are flat and flush with thealuminum surface. The height of the polished grains is thereforeapproximately 250 μm. The metal stub is fixed in a metal support holderand indented with a steel spherical indenter using an MTS universal testframe. The crosshead speed during the test is 2 μm/s. The steel ballused as the indenter is 3.2 mm in diameter. The maximum indentation loadis the same for all grains, and the load at first fracture is determinedfrom the load displacement curve as a load drop. After indentation, thegrains are imaged optically to document the existence of the cracks andthe crack pattern.

Using the first load drop as the pop-in load of the first ring crack,the Hertzian strength can be calculated. The Hertzian stress field iswell defined and axisymmetrical. The stresses are compressive rightunder the indenter and tensile outside a region defined by the radius ofthe contact area. At low loads, the field is completely elastic. For asphere of radius R and an applied normal load of P, the solutions forthe stress field are readily found following the original Hertzianassumption that the contact is friction free.

The radius of the contact area a is given by:

$\begin{matrix}{a^{3} = \frac{3{PR}}{4E^{*}}} & (1)\end{matrix}$

Where

$\begin{matrix}{E^{*} = \left( {\frac{1 - v_{1}^{2}}{E_{1}} + \frac{1 - v_{2}^{2}}{E_{2}}} \right)^{- 1}} & (2)\end{matrix}$

and E* is a combination of the Elastic modulus E and the Poisson's ratio□□ for the indenter and sample material, respectively.

The maximum contact pressure is given by:

$\begin{matrix}{P_{0} = {\left( \frac{3P}{2\pi a^{2}} \right) = \left( \frac{6PE^{*2}}{\pi^{3}R^{2}} \right)^{\frac{1}{3}}}} & (3)\end{matrix}$

The maximum shear stress is given by (assuming □=0.3): τ₁=0.31, p₀, atR=0 and z=0.48 a

The Hertzian strength is the maximum tensile stress at the onset ofcracking and is calculated according to: σ_(r)=⅓ (1-2□□) p₀ at R=a andz=0.

Using the first load drop as the load P in Eq. (3) the maximum tensilestress is calculated following the equation above, which is the value ofthe Hertzian strength for the specimen. In total, between 20 and 30individual shaped abrasive particle samples are tested for each grittype, and a range of Hertzian fracture stress is obtained. FollowingWeibull analysis procedures (as outlined in ASTM C1239), a Weibullprobability plot is generated, and the Weibull Characteristic strength(the scale value) and the Weibull modulus (the shape parameter) arecalculated for the distribution using the maximum likelihood procedure.

The shaped abrasive particles of the embodiments herein may have aparticular strength that facilitates formation of shaped abrasiveparticles with a particular 3SF. This may be achieved using any of thecompositions described in the embodiments herein, including but notlimited to, a single ceramic composition, a doped ceramic composition,or a composite composition. For example, the body 1801 of the shapedabrasive particle of the embodiments herein can have a strength within arange between at least about 350 MPa and not greater than about 1500MPa. For example, in one embodiment, the body 1801 can have a strengthof not greater than about 1490 MPa, such as not greater than about 1480MPa, not greater than about 1470 MPa, not greater than about 1460 MPa,not greater than about 1450 MPa, not greater than about 1440 MPa, notgreater than about 1430 MPa, not greater than about 1420 MPa, notgreater than about 1410 MPa, not greater than about 1400 MPa, notgreater than about 1390 MPa, not greater than about 1380 MPa, notgreater than about 1370 MPa, not greater than about 1360 MPa, notgreater than about 1350 MPa, not greater than about 1340 MPa, notgreater than about 1330 MPa, not greater than about 1320 MPa, notgreater than about 1310 MPa, not greater than about 1300 MPa, notgreater than about 1290 MPa, not greater than about 1280 MPa, notgreater than about 1270 MPa, not greater than about 1260 MPa, notgreater than about 1250 MPa, not greater than about 1240 MPa, notgreater than about 1230 MPa, not greater than about 1220 MPa, notgreater than about 1210 MPa, not greater than about 1200 MPa, notgreater than about 1190 MPa, not greater than about 1180 MPa, notgreater than about 1170 MPa, not greater than about 1160 MPa, notgreater than about 1150 MPa, not greater than about 1140 MPa, notgreater than about 1130 MPa, not greater than about 1120 MPa, notgreater than about 1110 MPa, not greater than about 1100 MPa, notgreater than about 1090 MPa, not greater than about 1080 MPa, notgreater than about 1070 MPa, not greater than about 1060 MPa, notgreater than about 1050 MPa, not greater than about 1040 MPa, notgreater than about 1030 MPa, not greater than about 1020 MPa, notgreater than about 1010 MPa, not greater than about 1000 MPa, notgreater than about 990 MPa, not greater than about 980 MPa, not greaterthan about 970 MPa, not greater than about 960 MPa, not greater thanabout 950 MPa, not greater than about 940 MPa, not greater than about930 MPa, not greater than about 920 MPa, not greater than about 910 MPa,not greater than about 900 MPa, not greater than about 890 MPa, notgreater than about 880 MPa, not greater than about 870 MPa, not greaterthan about 860 MPa, not greater than about 850 MPa, not greater thanabout 840 MPa, not greater than about 830 MPa, not greater than about820 MPa, not greater than about 810 MPa, not greater than about 800 MPa,not greater than about 790 MPa, not greater than about 780 MPa, notgreater than about 770 MPa, not greater than about 760 MPa, not greaterthan about 750 MPa, not greater than about 740 MPa, not greater thanabout 730 MPa, not greater than about 720 MPa, not greater than about710 MPa, not greater than about 700 MPa, not greater than about 690 MPa,not greater than about 680 MPa, not greater than about 670 MPa, notgreater than about 660 MPa, not greater than about 650 MPa, not greaterthan about 640 MPa, not greater than about 630 MPa, not greater thanabout 620 MPa, not greater than about 610 MPa, not greater than about600 MPa, not greater than about 590 MPa, not greater than about 580 MPa,not greater than about 570 MPa, not greater than about 560 MPa, notgreater than about 550 MPa, not greater than about 540 MPa, not greaterthan about 530 MPa, not greater than about 520 MPa, not greater thanabout 510 MPa, not greater than about 500 MPa, not greater than about490 MPa, not greater than about 480 MPa, not greater than about 470 MPa,not greater than about 460 MPa, not greater than about 450 MPa, notgreater than about 440 MPa, not greater than about 430 MPa, not greaterthan about 420 MPa, not greater than about 410 MPa, or even not greaterthan about 400 MPa. Still, in another non-limiting embodiment, the body1801 can have a strength of at least about 360 MPa, such as at leastabout 370 MPa, at least about 380 MPa, at least about 390 MPa, at leastabout 400 MPa, at least about 410 MPa, at least about 420 MPa, at leastabout 430 MPa, at least about 440 MPa, at least about 450 MPa, at leastabout 460 MPa, at least about 470 MPa, at least about 480 MPa, at leastabout 490 MPa, at least about 500 MPa, at least about 510 MPa, such asat least about 520 MPa, at least about 530 MPa, at least about 540 MPa,at least about 550 MPa, at least about 560 MPa, at least about 570 MPa,at least about 580 MPa, at least about 590 MPa, at least about 600 MPa,at least about 610 MPa, at least about 620 MPa, at least about 630 MPa,at least about 640 MPa, at least about 650 MPa, at least about 660 MPa,at least about 670 MPa, at least about 680 MPa, at least about 690 MPa,at least about 700 MPa, at least about 710 MPa, at least about 720 MPa,at least about 730 MPa, at least about 740 MPa, at least about 750 MPa,at least about 760 MPa, at least about 770 MPa, at least about 780 MPa,at least about 790 MPa, at least about 800 MPa, at least about 810 MPa,at least about 820 MPa, at least about 830 MPa, at least about 840 MPa,at least about 850 MPa, at least about 860 MPa, at least about 870 MPa,at least about 880 MPa, at least about 890 MPa, at least about 900 MPa,at least about 910 MPa, at least about 920 MPa, at least about 930 MPa,at least about 940 MPa, at least about 950 MPa, at least about 960 MPa,at least about 970 MPa, at least about 980 MPa, at least about 990 MPa,at least about 1000 MPa, at least about 1010 MPa, at least about 1020MPa, at least about 1030 MPa, at least about 1040 MPa, at least about1050 MPa, at least about 1060 MPa, at least about 1070 MPa, at leastabout 1080 MPa, at least about 1090 MPa, at least about 1100 MPa, atleast about 1110 MPa, at least about 1120 MPa, at least about 1130 MPa,at least about 1140 MPa, at least about 1150 MPa, at least about 1160MPa, at least about 1170 MPa, at least about 1180 MPa, at least about1190 MPa, at least about 1200 MPa, at least about 1210 MPa, at leastabout 1220 MPa, at least about 1230 MPa, at least about 1240 MPa, atleast about 1250 MPa, at least about 1260 MPa, at least about 1270 MPa,at least about 1280 MPa, at least about 1290 MPa, or even at least about1300 MPa. It will be appreciated that the strength of the body 1801 maybe within a range between any of the minimum and maximum values notedabove.

According to one aspect, empirical studies of shaped abrasive particleshave indicated that by controlling particular grain features of tipsharpness, strength, and Shape Index with respect to each other, thegrinding behavior (e.g., the self-sharpening behavior) of the shapedabrasive particles can be modified. Notably, the forming process can beundertaken in a manner such that the interrelationship of the grainfeatures of tip sharpness, Shape Index, and strength of the body areselected and controlled in a predetermined manner to influence thegrinding performance (e.g., self-sharpening behavior) of the shapedabrasive particle. For example, in one embodiment, the method of formingthe shaped abrasive particle can include selecting a material having apredetermined strength and forming the body of the shaped abrasiveparticle with a predetermined tip sharpness and predetermined ShapeIndex based upon the predetermined strength. That is, a material forforming the shaped abrasive particle may first be selected, such thatthe body will have a predetermined strength, and thereafter the grainfeatures of a predetermined tip sharpness and predetermined Shape Indexmay be selected and controlled based on the predetermined strength, suchthat the shaped abrasive particle may have improved performance overconventional shaped abrasive particles.

In still another embodiment, the method of forming the shaped abrasiveparticle can include selecting a material having a predetermined ShapeIndex and forming the body of the shaped abrasive particle with apredetermined tip sharpness and predetermined strength based upon thepredetermined Shape Index. That is, a shape of the body of the shapedabrasive particle may first be selected, and thereafter the grainfeatures of a predetermined tip sharpness and predetermined strength ofthe body may be selected and controlled based on the predetermined ShapeIndex, such that the shaped abrasive particle can have improvedperformance over conventional shaped abrasive particles.

In yet another approach, a method of forming a shaped abrasive particlecan include selecting a predetermined tip sharpness of a body of theshaped abrasive particle. After predetermining the tip sharpness of thebody, the Shape Index and the strength of the body may be selected andcontrolled based upon the predetermined tip sharpness. Such a processmay facilitate formation of a shaped abrasive particle having improvedperformance over conventional shaped abrasive particles.

In yet another embodiment, the method of forming the shaped abrasiveparticle can include selecting a material having a predetermined height,which may be an average height, an interior height, or height at an edgeor tip of the body, and forming the body of the shaped abrasive particlewith a predetermined tip sharpness, predetermined strength, andpredetermined Shape Index based on the predetermined height. That is, aheight of the body of the shaped abrasive particle may first beselected, and thereafter the grain features of a predetermined tipsharpness, strength, and Shape Index of the body may be selected andcontrolled based on the predetermined height, such that the shapedabrasive particle can have improved performance over conventional shapedabrasive particles.

Moreover, through empirical studies, it has been found that theperformance of the shaped abrasive particle may be initially predictedby the interrelationship of the tip sharpness, strength, and ShapeIndex, which may be evaluated based upon a sharpness-shape-strengthfactor (3SF) according to the formula: 3SF=[(S*R*B²)/2500], wherein “S”represents the strength of the body (in MPa), R represents the tipsharpness of the body (in microns), and “B” represents the Shape Indexof the body. The 3SF formula is intended to provide an initialprediction of the effectiveness of grinding behavior of the particlebased upon the interrelationship of the grain features. It should benoted that other factors, such as aspects of the abrasive article inwhich the shaped abrasive particle is integrated, may also influence thebehavior of the particle.

In accordance with one embodiment, the body of the shaped abrasiveparticle may have a particular 3SF value within a range between at leastabout 0.7 and not greater than about 1.7. In at least one embodiment,the body can have a 3SF of at least about 0.72, such as at least about0.75, at least about 0.78, at least about 0.8, at least about 0.82, atleast about 0.85, at least about 0.88, at least about 0.90, at leastabout 0.92, at least about 0.95, or even at least about 0.98. In yetanother instance, the body can have a 3SF of not greater than about1.68, such as not greater than about 1.65, not greater than about 1.62,not greater than about 1.6, not greater than about 1.58, not greaterthan about 1.55, not greater than about 1.52, not greater than about1.5, not greater than about 1.48, not greater than about 1.45, notgreater than about 1.42, not greater than about 1.4, not greater thanabout 1.38, not greater than about 1.35, not greater than about 1.32,not greater than about 1.3, not greater than about 1.28, not greaterthan about 1.25, not greater than about 1.22, not greater than about1.2, not greater than about 1.18, not greater than about 1.15, notgreater than about 1.12, not greater than about 1.1. It will beappreciated that the body can have a 3SF value within a range betweenany of the minimum and maximum values noted above.

In addition to the foregoing grain features and 3SF values of theembodiments herein, in certain instances, the height of the grain may bean additional or alternative grain feature that may be interrelated tocertain grain features described herein. In particular, the height ofthe grain may be controlled with respect to any of the grain features(e.g., strength and tip sharpness) to facilitate improved grindingperformance of the shaped abrasive particles and abrasive articles usingsuch shaped abrasive particles. Notably, the shaped abrasive particlesof the embodiments herein can have a particular height, which may beinterrelated to certain grain features, such that stresses encounteredduring grinding may be distributed throughout the body in a manner tofacilitate improved self-sharpening behavior. According to oneembodiment, the body of the shaped abrasive particles can have a height(h) within a range between about 70 microns and about 500 microns, suchas within a range between about 175 microns to about 350 microns, suchas between about 175 microns and about 300 microns, or even within arange between about 200 microns and about 300 microns.

The shaped abrasive particles of the embodiments herein having theparticular grain features and 3SF can have any of the other features ofthe embodiments described herein. In one aspect, the body 1701 of theshaped abrasive particle can have a particular composition. For example,the body 1701 may include a ceramic material, such as a polycrystallineceramic material, and more particularly an oxide. The oxide may include,for example alumina. In certain instances, the body may include amajority content of alumina, such as at least about 95 wt % alumina forthe total weight of the body, or such as at least about 95.1 wt %, atleast about 95.2 wt %, at least about 95.3 wt %, at least about 95.4 wt%, at least about 95.5 wt %, at least about 95.6 wt %, at least about95.7 wt %, at least about 95.8 wt %, at least about 95.9 wt %, at leastabout 96 wt %, at least about 96.1 wt %, at least about 96.2 wt %, atleast about 96.3 wt %, at least about 96.4 wt %, at least about 96.5 wt%, at least about 96.6 wt %, at least about 96.7 wt %, at least about96.8 wt %, at least about 96.9 wt %, at least about 97 wt %, at leastabout 97.1 wt %, at least about 97.2 wt %, at least about 975.3 wt %, atleast about 97.4 wt %, or even at least about 97.5 wt % alumina for thetotal weight of the body. Still, in another non-limiting embodiment, thebody 1701 may include a content of alumina not greater than about 99.5wt %, such as not greater than about 99.4 wt %, not greater than about99.3 wt %, not greater than about 99.2 wt %, not greater than about 99.1wt %, not greater than about 99 wt %, not greater than about 98.9 wt %,not greater than about 98.8 wt %, not greater than about 98.7 wt %, notgreater than about 98.6 wt %, not greater than about 98.5 wt %, notgreater than about 98.4 wt %, not greater than about 98.3 wt %, notgreater than about 98.2 wt %, not greater than about 98.1 wt %, notgreater than about 98 wt %, not greater than about 97.9 wt %, notgreater than about 97.8 wt %, not greater than about 97.7 wt %, notgreater than about 97.6 wt %, or even not greater than about 97.5 wt %alumina for the total weight of the body 1201. It will be appreciatedthat the body 1701 may include a content of alumina within a rangebetween any of the minimum and maximum values noted above. Moreover, inat least one embodiment, the body 1701 may consist essentially ofalumina.

In a particular instance, the shaped abrasive particles of theembodiments herein can have a particular draft angle at the intersectionof the smallest major surface and the side surface, which may beindicative of a particular aspect of forming and/or may facilitateimproved performance of the abrasive particle. In one particularinstance, the shaped abrasive particles herein can have an average draftangle, which can be an average measure of draft angle for astatistically relevant and random sample size of shaped abrasiveparticles (e.g., at least 20 particles). In a particular instance, theaverage draft angle can be not greater than 95°, such as not greaterthan 94° or no greater than 93° or not greater than 92° or not greaterthan 91° or even not greater than 90°. In at least one non-limitingembodiment, the shaped abrasive particles of the embodiments herein canhave an average draft angle of at least 80° such as at least 82° or atleast 84° or at least 85° or at least 86° or at least 87°. It will beappreciated that the shaped abrasive particles of the embodiments hereincan have an average draft angle within a range including any of theminimum and maximum values noted above, including but not limited to,within a range of at least 80° and not greater than 95° or within arange including at least 80° and not greater than 94° or within a rangeincluding at least 82° and not greater than 93° or within a rangeincluding at least 84° and not greater than 93°.

The draft angle can be measured by cutting the shaped abrasive particlein half at an approximately 90° angle with respect to the major surfaceand at a perpendicular angle to one of the side surfaces, such as shownby the dotted line in FIG. 18C. As best as possible, the sectioning lineshould extend perpendicular to the side surface and through the midpointof a major surface of the particle. The portion of the shaped abrasiveparticle is then mounted and viewed via SEM in a manner that is similarto that provided in FIG. 18D. A suitable program for such includesImageJ software. Using the image of the body, the smallest major surfaceis determined by identifying the largest major surface and selecting thesurface opposite thereof. Certain shaped abrasive particles may have agenerally square cross-sectional shape. To identify the smallest majorsurface, the largest major surface must first be determined. Thesmallest major surface is that surface opposite the largest majorsurface. The imaging software, such as ImageJ may be utilized to assistwith the determination of the smallest major surface. Using a suitableimage processing software (e.g., ImageJ) draw a straight line along bothof the major surfaces between the corners adjoining the major surfacesand the sidewall as provided by the lines below in FIG. 18D. Using theimage analysis software, measure the line that longer. The shorter ofthe two lines is presumed to be the smaller of the two major surfaces.In the case provided in FIG. 18D, the line on the right of the image isshorter and the draft angle should be measured at the corner identifiedat the upper right-hand corner, which is also illustrated in FIG. 18E.

To measure the draft angle, lines can be drawn along the smallest majorsurface and the side surface to form an intersecting angle as providedin FIG. 18E. The lines are drawn taking into consideration the shape ofthe surfaces as a whole and ignoring imperfections or othernon-representative surface undulations at the corner of the particle(e.g., cracks or chips due to mounting procedures, etc.). Moreover, theline representing the smaller major surface is drawn to represent theportion of the major surface that connects the sidewall at the draftangle. The draft angle (i.e., the angle of the body as measured at theintersection) is determined by the interior angle formed at theintersection of the lines.

As noted in embodiments herein, the body of the shaped abrasiveparticles may be formed to include certain additives. The additives canbe non-organic species, including but not limited to an oxide, a metalelement, a rare-earth element, and a combination thereof. In oneparticular instance, the additive may be a dopant material, which may bepresent in a particular minor amount sufficient to affect themicrostructure of the material, but not necessarily present in a traceamount or less. The dopant material may include an element selected fromthe group consisting of an alkali element, an alkaline earth element, arare earth element, a transition metal element, and a combinationthereof. More particularly, the dopant material can be an elementselected from the group consisting of hafnium, zirconium, niobium,tantalum, molybdenum, vanadium, lithium, sodium, potassium, magnesium,calcium, strontium, barium, scandium, yttrium, lanthanum, cesium,praseodymium, chromium, cobalt, iron, germanium, manganese, nickel,titanium, zinc, and a combination thereof. In still a more particularembodiment, the dopant material may include a magnesium-containingspecies, including but not limited to, magnesium oxide (MgO).

Certain compositions of the shaped abrasive particles of the embodimentsherein can include a particular content of magnesium oxide. For example,the body 1701 may include a content of the magnesium-containing speciesof at least about 0.5 wt %, such as at least about 0.6 wt %, at leastabout 0.7 wt %, at least about 0.8 wt %, at least about 0.9 wt %, atleast about 1 wt %, at least about 1.1 wt %, at least about 1.2 wt %, atleast about 1.3 wt %, at least about 1.4 wt %, at least about 1.5 wt %,at least about 1.6 wt %, at least about 1.7 wt %, at least about 1.8 wt%, at least about 1.9 wt %, at least about 2 wt %, at least about 2.1 wt%, at least about 2.2 wt %, at least about 2.3 wt %, at least about 2.4wt %, or even at least about 2.5 wt % for the total weight of the body1701. In still another non-limiting embodiment, the body 1701 mayinclude a content of the magnesium-containing species of not greaterthan about 8 wt %, not greater than about 7 wt %, not greater than about6 wt %, not greater than about 5 wt %, not greater than about 4.9 wt %,not greater than about 4.8 wt %, not greater than about 4.7 wt %, notgreater than about 4.6 wt %, not greater than about 4.5 wt %, notgreater than about 4.4 wt %, not greater than about 4.3 wt %, notgreater than about 4.2 wt %, not greater than about 4.1 wt %, notgreater than about 4 wt %, not greater than about 3.9 wt %, not greaterthan about 3.8 wt %, not greater than about 3.7 wt %, not greater thanabout 3.6 wt %, not greater than about 3.5 wt %, not greater than about3.4 wt %, not greater than about 3.3 wt %, not greater than about 3.2 wt%, not greater than about 3.1 wt %, not greater than about 3 wt %, notgreater than about 2.9 wt %, not greater than about 2.8 wt %, notgreater than about 2.7 wt %, not greater than about 2.6 wt %, notgreater than about 2.5 wt %. It will be appreciated that the content ofthe magnesium-containing species within the body may be within a rangebetween any of the minimum and maximum values noted above. Furthermore,in at least one embodiment, the body 1701 may consist essentially ofalumina (Al₂O₃) and the magnesium-containing species (e.g., MgO and/or amagnesium aluminate).

Moreover, as noted herein, the body of a shaped abrasive particle of anyof the embodiments herein may be formed of a polycrystalline materialincluding grains, which may be made of materials such as nitrides,oxides, carbides, borides, oxynitrides, diamond, and a combinationthereof. Further, the body 1701 can be essentially free of an organicmaterial, essentially free of rare earth elements, and essentially freeof iron. Being essentially free is understood to mean that the body isformed in a manner to exclude such materials, but the body may notnecessarily be completely free of such materials as they may be presentin trace amounts or less.

A Fixed Abrasive Article

After forming or sourcing the shaped abrasive particles, the particlescan be combined with other materials to form a fixed abrasive article.In a fixed abrasive, the shaped abrasive particles can be coupled to amatrix or substrate and used for material removal operations. Somesuitable exemplary fixed abrasive articles can include bonded abrasivearticles wherein the shaped abrasive particles are contained in a threedimensional matrix of bond material. In other instances, the fixedabrasive article may be a coated abrasive article, wherein the shapedabrasive particles may be dispersed in a single layer overlying abacking (e.g., a substrate) and bonded to the backing using one or moreadhesive layers.

FIG. 5A includes an illustration of a bonded abrasive articleincorporating the abrasive particulate material in accordance with anembodiment. As illustrated, the bonded abrasive 590 can include a bondmaterial 591, abrasive particulate material 592 contained in the bondmaterial, and porosity 598 within the bond material 591. In particularinstances, the bond material 591 can include an organic material,inorganic material, and a combination thereof. Suitable organicmaterials can include polymers, such as epoxies, resins, thermosets,thermoplastics, polyimides, polyamides, and a combination thereof.Certain suitable inorganic materials can include metals, metal alloys,vitreous phase materials, crystalline phase materials, ceramics, and acombination thereof.

In some instances, the abrasive particulate material 592 of the bondedabrasive 590 can include shaped abrasive particles 593, 594, 595, and596. In particular instances, the shaped abrasive particles 593, 594,595, and 596 can be different types of particles, which can differ fromeach other in composition, two-dimensional shape, three-dimensionalshape, size, and a combination thereof as described in the embodimentsherein. Alternatively, the bonded abrasive article can include a singletype of shaped abrasive particle.

The bonded abrasive 590 can include a type of abrasive particulatematerial 597 representing diluent abrasive particles, which can differfrom the shaped abrasive particles 593, 594, 595, and 596 incomposition, two-dimensional shape, three-dimensional shape, size, and acombination thereof.

The porosity 598 of the bonded abrasive 590 can be open porosity, closedporosity, and a combination thereof. The porosity 598 may be present ina majority amount (vol %) based on the total volume of the body of thebonded abrasive 590. Alternatively, the porosity 598 can be present in aminor amount (vol %) based on the total volume of the body of the bondedabrasive 590. The bond material 591 may be present in a majority amount(vol %) based on the total volume of the body of the bonded abrasive590. Alternatively, the bond material 591 can be present in a minoramount (vol %) based on the total volume of the body of the bondedabrasive 590. Additionally, abrasive particulate material 592 can bepresent in a majority amount (vol %) based on the total volume of thebody of the bonded abrasive 590. Alternatively, the abrasive particulatematerial 592 can be present in a minor amount (vol %) based on the totalvolume of the body of the bonded abrasive 590.

FIG. 5B includes a cross-sectional illustration of a coated abrasivearticle in accordance with an embodiment. In particular, the coatedabrasive article 500 can include a substrate 501 (e.g., a backing) andat least one adhesive layer overlying a surface of the substrate 501.The adhesive layer can include a make coat 503 and/or a size coat 504.The coated abrasive article 500 can include abrasive particulatematerial 510, which can include shaped abrasive particles 505 of any ofthe embodiments herein and a second type of abrasive particulatematerial 507 in the form of diluent abrasive particles having a randomshape, which may not necessarily be shaped abrasive particles. Theshaped abrasive particles 505 of FIG. 5B are illustrated generally forpurposes or discussion, and it will be appreciated that the coatedabrasive article can include any shaped abrasive particles of theembodiments herein. The make coat 503 can be overlying the surface ofthe substrate 501 and surrounding at least a portion of the shapedabrasive particles 505 and second type of abrasive particulate material507. The size coat 504 can be overlying and bonded to the shapedabrasive particles 505 and second type of abrasive particulate material507 and the make coat 503.

According to one embodiment, the substrate 501 can include an organicmaterial, inorganic material, and a combination thereof. In certaininstances, the substrate 501 can include a woven material. However, thesubstrate 501 may be made of a non-woven material. Particularly suitablesubstrate materials can include organic materials, including polymerssuch as polyester, polyurethane, polypropylene, and/or polyimides suchas KAPTON from DuPont, and paper. Some suitable inorganic materials caninclude metals, metal alloys, and particularly, foils of copper,aluminum, steel, and a combination thereof. The backing can include oneor more additives selected from the group of catalysts, coupling agents,curants, anti-static agents, suspending agents, anti-loading agents,lubricants, wetting agents, dyes, fillers, viscosity modifiers,dispersants, defoamers, and grinding agents.

A polymer formulation may be used to form any of a variety of layers ofthe coated abrasive article 500 such as, for example, a frontfill, apre-size, the make coat, the size coat, and/or a supersize coat. Whenused to form the frontfill, the polymer formulation generally includes apolymer resin, fibrillated fibers (preferably in the form of pulp),filler material, and other optional additives. Suitable formulations forsome frontfill embodiments can include material such as a phenolicresin, wollastonite filler, defoamer, surfactant, a fibrillated fiber,and a balance of water. Suitable polymeric resin materials includecurable resins selected from thermally curable resins including phenolicresins, urea/formaldehyde resins, phenolic/latex resins, as well ascombinations of such resins. Other suitable polymeric resin materialsmay also include radiation curable resins, such as those resins curableusing electron beam, UV radiation, or visible light, such as epoxyresins, acrylated oligomers of acrylated epoxy resins, polyester resins,acrylated urethanes and polyester acrylates and acrylated monomersincluding monoacrylated, multiacrylated monomers. The formulation canalso comprise a nonreactive thermoplastic resin binder which can enhancethe self-sharpening characteristics of the deposited abrasive particlesby enhancing the erodability. Examples of such thermoplastic resininclude polypropylene glycol, polyethylene glycol, andpolyoxypropylene-polyoxyethene block copolymer, etc. Use of a frontfillon the substrate 501 can improve the uniformity of the surface, forsuitable application of the make coat 503 and improved application andorientation of shaped abrasive particles 505 in a predeterminedorientation.

The make coat 503 can be applied to the surface of the substrate 501 ina single process, or alternatively, the abrasive particulate material510 can be combined with a make coat 503 material and applied as amixture to the surface of the substrate 501. Suitable materials of themake coat 503 can include organic materials, particularly polymericmaterials, including for example, polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, polyvinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and mixtures thereof.In one embodiment, the make coat 503 can include a polyester resin. Thecoated substrate can then be heated in order to cure the resin and theabrasive particulate material to the substrate. In general, the coatedsubstrate 501 can be heated to a temperature of between about 100° C. toless than about 250° C. during this curing process.

The abrasive particulate material 510 can include shaped abrasiveparticles 505 according to embodiments herein. In particular instances,the abrasive particulate material 510 may include different types ofshaped abrasive particles 505. The different types of shaped abrasiveparticles can differ from each other in composition, in two-dimensionalshape, in three-dimensional shape, in size, and a combination thereof asdescribed in the embodiments herein. As illustrated, the coated abrasive500 can include a shaped abrasive particle 505 which may have any of theshapes of the shaped abrasive particles of the embodiments herein.

The other type of abrasive particles 507 can be diluent particlesdifferent than the shaped abrasive particles 505. For example, thediluent particles can differ from the shaped abrasive particles 505 incomposition, in two-dimensional shape, in three-dimensional shape, insize, and a combination thereof. For example, the abrasive particles 507can represent conventional, crushed abrasive grit having random shapes.The abrasive particles 507 may have a median particle size less than themedian particle size of the shaped abrasive particles 505.

After sufficiently forming the make coat 503 with the abrasiveparticulate material 510, the size coat 504 can be formed to overlie andbond the abrasive particulate material 510 in place. The size coat 504can include an organic material, may be made essentially of a polymericmaterial, and notably, can use polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, poly vinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and mixtures thereof.

According to one embodiment, the shaped abrasive particles 505 can beoriented in a predetermined orientation relative to each other and/orthe substrate 501. While not completely understood, it is thought thatone or a combination of dimensional features may be responsible forimproved orientation of the shaped abrasive particles 505. According toone embodiment, the shaped abrasive particles 505 can be oriented in aflat orientation relative to the substrate 501, such as that shown inFIG. 5B. In the flat orientation, the bottom surface 304 of the shapedabrasive particles can be closest to a surface of the substrate 501 andthe upper surface 303 of the shaped abrasive particles 505 can bedirected away from the substrate 501 and configured to conduct initialengagement with a workpiece.

According to another embodiment, the shaped abrasive particles 505 canbe placed on a substrate 501 in a predetermined side orientation, suchas that shown in FIG. 6. In particular instances, a majority of theshaped abrasive particles 505 of the total content of shaped abrasiveparticles 505 on the abrasive article 500 can have a predetermined sideorientation. In the side orientation, the bottom surface 304 of theshaped abrasive particles 505 can be spaced away from and angledrelative to the surface of the substrate 501. In particular instances,the bottom surface 304 can form an obtuse angle (B) relative to thesurface of the substrate 501. Moreover, the upper surface 303 is spacedaway and angled relative to the surface of the substrate 501, which inparticular instances, may define a generally acute angle (A). In a sideorientation, a side surface 305 can be closest to the surface of thesubstrate 501, and more particularly, may be in direct contact with asurface of the substrate 501.

For certain other abrasive articles herein, at least about 55% of theplurality of shaped abrasive particles 505 on the abrasive article 500can be coupled to the backing in a predetermined side orientation.Still, the percentage may be greater, such as at least about 60%, atleast about 65%, at least about 70%, at least about 75%, at least about77%, at least about 80%, at least about 81%, or even at least about 82%.And for one non-limiting embodiment, an abrasive article 500 may beformed using the shaped abrasive particles 505 herein, wherein notgreater than about 99% of the total content of shaped abrasive particleshave a predetermined side orientation.

To determine the percentage of particles in a predetermined orientation,a 2D microfocus x-ray image of the abrasive article 500 is obtainedusing a CT scan machine run in the conditions of Table 1 below. TheX-ray 2D imaging is conducted on shaped abrasive particles on a backingwith Quality Assurance software. A specimen mounting fixture utilizes aplastic frame with a 4″×4″ window and an 00.5″ solid metallic rod, thetop part of which is half flattened with two screws to fix the frame.Prior to imaging, a specimen is clipped over one side of the frame wherethe screw heads face the incidence direction of the X-rays. Then fiveregions within the 4″×4″ window area are selected for imaging at 120kV/80 μA. Each 2D projection is recorded with the X-ray off-set/gaincorrections and at a magnification of 15 times.

TABLE 1 Field of view Voltage Current per image (kV) (μA) Magnification(mm × mm) Exposure time 120 80 15X 16.2 × 13.0 500 ms/2.0 fps

The image is then imported and analyzed using the ImageJ program,wherein different orientations are assigned values according to Table 2below. FIG. 11 includes images representative of portions of a coatedabrasive article according to an embodiment, which images can be used toanalyze the orientation of shaped abrasive particles on the backing.

TABLE 2 Cell marker type Comments 1 Grains on the perimeter of theimage, partially exposed - standing up 2 Grains on the perimeter of theimage, partially exposed - down 3 Grains on the image, completelyexposed - standing vertical 4 Grains on the image, completely exposed -down 5 Grains on the image, completely exposed - standing slanted(between standing vertical and down)

Three calculations are then performed as provided below in Table 3.After conducting the calculations, the percentage of grains in aparticular orientation (e.g., side orientation) per square centimetercan be derived.

TABLE 3 5) Parameter Protocol* % grains up ((0.5 × 1) + 3 + 5)/(1 + 2 +3 + 4 + 5) Total # of grains per (1 + 2 + 3 + 4 + 5) cm² # of grains upper (% grains up × Total # of grains per cm² cm²

-   -   These are all normalized with respect to the representative area        of the image.    -   A scale factor of 0.5 was applied to account for the fact that        they are not completely present in the image.

Furthermore, the abrasive articles made with the shaped abrasiveparticles can utilize various contents of the shaped abrasive particles.For example, the abrasive articles can be coated abrasive articlesincluding a single layer of a plurality of shaped abrasive particles inan open-coat configuration or a closed-coat configuration. For example,the plurality of shaped abrasive particles can define an open-coatabrasive article having a coating density of shaped abrasive particlesof not greater than about 70 particles/cm². In other instances, theopen-coat density of shaped abrasive particles per square centimeter ofabrasive article may be not greater than about 65 particles/cm², such asnot greater than about 60 particles/cm², not greater than about 55particles/cm², or even not greater than about 50 particles/cm². Still,in one non-limiting embodiment, the density of the open-coat abrasivearticle using the shaped abrasive particle herein can be at least about5 particles/cm², or even at least about 10 particles/cm². It will beappreciated that the open-coat density of the coated abrasive articlecan be within a range between any of the above minimum and maximumvalues.

In an alternative embodiment, the plurality of shaped abrasive particlescan define a closed-coat abrasive article having a coating density ofshaped abrasive particles of at least about 75 particles/cm², such as atleast about 80 particles/cm², at least about 85 particles/cm², at leastabout 90 particles/cm², at least about 100 particles/cm². Still, in onenon-limiting embodiment, the closed-coat density of the coated abrasivearticle using the shaped abrasive particle herein can be not greaterthan about 500 particles/cm². It will be appreciated that the closedcoat density of the coated abrasive article can be within a rangebetween any of the above minimum and maximum values.

In certain instances, the abrasive article can have an open-coat densityof a coating not greater than about 50% of abrasive particulate materialcovering the exterior abrasive surface of the article. In otherembodiments, the percentage coating of the abrasive particulate materialrelative to the total area of the abrasive surface can be not greaterthan about 40%, not greater than about 30%, not greater than about 25%,or even not greater than about 20%. Still, in one non-limitingembodiment, the percentage coating of the abrasive particulate materialrelative to the total area of the abrasive surface can be at least about5%, such as at least about 10%, at least about 15%, at least about 20%,at least about 25%, at least about 30%, at least about 35%, or even atleast about 40%. It will be appreciated that the percent coverage ofshaped abrasive particles for the total area of abrasive surface can bewithin a range between any of the above minimum and maximum values.

Some abrasive articles may have a particular content of abrasiveparticles for a length (e.g., ream) of the backing or the substrate 501.For example, in one embodiment, the abrasive article may utilize anormalized weight of shaped abrasive particles of at least about 20lbs/ream, such as at least about 25 lbs/ream, or even at least about 30lbs/ream. Still, in one non-limiting embodiment, the abrasive articlescan include a normalized weight of shaped abrasive particles of notgreater than about 60 lbs/ream, such as not greater than about 50lbs/ream, or even not greater than about 45 lbs/ream. It will beappreciated that the abrasive articles of the embodiments herein canutilize a normalized weight of shaped abrasive particles within a rangebetween any of the above minimum and maximum values.

The plurality of shaped abrasive particles on an abrasive article asdescribed herein can define a first portion of a batch of abrasiveparticles, and the features described in the embodiments herein canrepresent features that are present in at least a first portion of abatch of shaped abrasive particles. Moreover, according to anembodiment, control of one or more process parameters as alreadydescribed herein also can control the prevalence of one or more featuresof the shaped abrasive particles of the embodiments herein. Theprovision of one or more features of any shaped abrasive particle of abatch may facilitate alternative or improved deployment of the particlesin an abrasive article and may further facilitate improved performanceor use of the abrasive article. The batch may also include a secondportion of abrasive particles. The second portion of abrasive particlescan include diluent particles.

In accordance with one aspect of the embodiments herein, a fixedabrasive article can include a blend of abrasive particles. The blend ofabrasive particles can include a first type of shaped abrasive particleand a second type of shaped abrasive particle. The first type of shapedabrasive particle can include any features of the shaped abrasiveparticles of the embodiments herein. The second type of shaped abrasiveparticle can include any features of the shaped abrasive particles ofthe embodiments herein. Moreover, it will be appreciated in light of thepresent disclosure that one or more different types of abrasiveparticles, including abrasive particles of the embodiments herein and/orconventional abrasive particles may be combined in a fixed abrasive toimprove the overall performance of the abrasive article. This mayinclude the use of blends of different types of abrasive particles,wherein the different types of abrasive particles may differ in size,shape, hardness, fracture toughness, strength, tip sharpness, ShapeIndex, composition, type and/or content of dopants, and a combinationthereof.

The blend of abrasive particles can include a first type of shapedabrasive particle present in a first content (C1), which may beexpressed as a percentage (e.g., a weight percent) of the first type ofshaped abrasive particles as compared to the total content of particlesof the blend. Furthermore, the blend of abrasive particles may include asecond content (C2) of the second type of shaped abrasive particles,expressed as a percentage (e.g., a weight percent) of the second type ofshaped abrasive particles relative to the total weight of the blend. Thefirst content can be the same as or different from the second content.For example, in certain instances, the blend can be formed such that thefirst content (C1) can be not greater than about 90% of the totalcontent of the blend. In another embodiment, the first content may beless, such as not greater than about 85%, not greater than about 80%,not greater than about 75%, not greater than about 70%, not greater thanabout 65%, not greater than about 60%, not greater than about 55%, notgreater than about 50%, not greater than about 45%, not greater thanabout 40%, not greater than about 35%, not greater than about 30%, notgreater than about 25%, not greater than about 20%, not greater thanabout 15%, not greater than about 10%, or even not greater than about5%. Still, in one non-limiting embodiment, the first content of thefirst type of shaped abrasive particles may be present in at least about1% of the total content of abrasive particles of the blend. In yet otherinstances, the first content (C1) may be at least about 5%, such as atleast about 10%, at least about 15%, at least about 20%, at least about25%, at least about 30%, at least about 35%, at least about 40%, atleast about 45%, at least about 50%, at least about 55%, at least about60%, at least about 65%, at least about 70%, at least about 75%, atleast about 80%, at least about 85%, at least about 90%, or even atleast about 95%. It will be appreciated that the first content (C1) maybe present within a range between any of the minimum and maximumpercentages noted above.

The blend of abrasive particles may include a particular content of thesecond type of shaped abrasive particle. For example, the second content(C2) may be not greater than about 98% of the total content of theblend. In other embodiments, the second content may be not greater thanabout 95%, such as not greater than about 90%, not greater than about85%, not greater than about 80%, not greater than about 75%, not greaterthan about 70%, not greater than about 65%, not greater than about 60%,not greater than about 55%, not greater than about 50%, not greater thanabout 45%, not greater than about 40%, not greater than about 35%, notgreater than about 30%, not greater than about 25%, not greater thanabout 20%, not greater than about 15%, not greater than about 10%, oreven not greater than about 5%. Still, in one non-limiting embodiment,the second content (C2) may be present in an amount of at least about 1%of the total content of the blend. For example, the second content maybe at least about 5%, such as at least about 10%, at least about 15%, atleast about 20%, at least about 25%, at least about 30%, at least about35%, at least about 40%, at least about 45%, at least about 50%, atleast about 55%, at least about 60%, at least about 65%, at least about70%, at least about 75%, at least about 80%, at least about 85%, atleast about 90%, or even at least about 95%. It will be appreciated thatthe second content (C2) can be within a range between any of the minimumand maximum percentages noted above.

In accordance with another embodiment, the blend of abrasive particlesmay have a blend ratio (C1/C2) that may define a ratio between the firstcontent (C1) and the second content (C2). For example, in oneembodiment, the blend ratio (C1/C2) may be not greater than about 10. Inyet another embodiment, the blend ratio (C1/C2) may be not greater thanabout 8, such as not greater than about 6, not greater than about 5, notgreater than about 4, not greater than about 3, not greater than about2, not greater than about 1.8, not greater than about 1.5, not greaterthan about 1.2, not greater than about 1, not greater than about 0.9,not greater than about 0.8, not greater than about 0.7, not greater thanabout 0.6, not greater than about 0.5, not greater than about 0.4, notgreater than about 0.3, or even not greater than about 0.2. Still, inanother non-limiting embodiment, the blend ratio (C1/C2) may be at leastabout 0.1, such as at least about 0.15, at least about 0.2, at leastabout 0.22, at least about 0.25, at least about 0.28, at least about0.3, at least about 0.32, at least about 0.3, at least about 0.4, atleast about 0.45, at least about 0.5, at least about 0.55, at leastabout 0.6, at least about 0.65, at least about 0.7, at least about 0.75,at least about 0.8, at least about 0.9, at least about 0.95, at leastabout 1, at least about 1.5, at least about 2, at least about 3, atleast about 4, or even at least about 5. It will be appreciated that theblend ratio (C1/C2) may be within a range between any of the minimum andmaximum values noted above.

In at least one embodiment, the blend of abrasive particles can includea majority content of shaped abrasive particles. That is, the blend canbe formed primarily of shaped abrasive particles, including, but notlimited to, a first type of shaped abrasive particle and a second typeof shaped abrasive particle. In at least one particular embodiment, theblend of abrasive particles can consist essentially of the first type ofshaped abrasive particle and the second type of shaped abrasiveparticle. However, in other non-limiting embodiments, the blend mayinclude other types of abrasive particles. For example, the blend mayinclude a third type of abrasive particle that may include aconventional abrasive particle or a shaped abrasive particle. The thirdtype of abrasive particle may include a diluent type of abrasiveparticle having an irregular shape, which may be achieved throughconventional crushing and comminution techniques.

According to another embodiment, the blend of abrasive particles caninclude a plurality of shaped abrasive particles and each of the shapedabrasive particles of the plurality may be arranged in a controlledorientation relative to a backing, such as a substrate of a coatedabrasive article. Suitable exemplary controlled orientations can includeat least one of a predetermined rotational orientation, a predeterminedlateral orientation, and a predetermined longitudinal orientation. In atleast one embodiment, the plurality of shaped abrasive particles havinga controlled orientation can include at least a portion of the firsttype of shaped abrasive particles of the blend, at least a portion ofthe second type of shaped abrasive particles of the blend, and acombination thereof. More particularly, the plurality of shaped abrasiveparticles having a controlled orientation can include all of the firsttype of shaped abrasive particles. In still another embodiment, theplurality of shaped abrasive particles arranged in a controlledorientation relative to the backing may include all of the second typeof shaped abrasive particles within the blend of abrasive particles.

FIG. 7 includes a top view illustration of a portion of a coatedabrasive article including shaped abrasive particles having controlledorientation. As illustrated, the coated abrasive article 700 includes abacking 701 that can be defined by a longitudinal axis 780 that extendsalong and defines a length of the backing 701 and a lateral axis 781that extends along and defines a width of the backing 701. In accordancewith an embodiment, a shaped abrasive particle 702 can be located in afirst, predetermined position 712 defined by a particular first lateralposition relative to the lateral axis of 781 of the backing 701 and afirst longitudinal position relative to the longitudinal axis 780 of thebacking 701. Furthermore, a shaped abrasive particle 703 may have asecond, predetermined position 713 defined by a second lateral positionrelative to the lateral axis 781 of the backing 701, and a firstlongitudinal position relative to the longitudinal axis 780 of thebacking 701 that is substantially the same as the first longitudinalposition of the shaped abrasive particle 702. Notably, the shapedabrasive particles 702 and 703 may be spaced apart from each other by alateral space 721, defined as a smallest distance between the twoadjacent shaped abrasive particles 702 and 703 as measured along alateral plane 784 parallel to the lateral axis 781 of the backing 701.In accordance with an embodiment, the lateral space 721 can be greaterthan zero, such that some distance exists between the shaped abrasiveparticles 702 and 703. However, while not illustrated, it will beappreciated that the lateral space 721 can be zero, allowing for contactand even overlap between portions of adjacent shaped abrasive particles.

As further illustrated, the coated abrasive article 700 can include ashaped abrasive particle 704 located at a third, predetermined position714 defined by a second longitudinal position relative to thelongitudinal axis 780 of the backing 701 and also defined by a thirdlateral position relative to a lateral plane 785 parallel to the lateralaxis 781 of the backing 701 and spaced apart from the lateral axis 784.Further, as illustrated, a longitudinal space 723 may exist between theshaped abrasive particles 702 and 704, which can be defined as asmallest distance between the two adjacent shaped abrasive particles 702and 704 as measured in a direction parallel to the longitudinal axis780. In accordance with an embodiment, the longitudinal space 723 can begreater than zero. Still, while not illustrated, it will be appreciatedthat the longitudinal space 723 can be zero, such that the adjacentshaped abrasive particles are touching, or even overlapping each other.

FIG. 8A includes a top view illustration of a portion of an abrasivearticle including shaped abrasive particles in accordance with anembodiment. As illustrated, the abrasive article 800 can include ashaped abrasive particle 802 overlying a backing 801 in a first positionhaving a first rotational orientation relative to a lateral axis 781defining the width of the backing 801. In particular, the shapedabrasive particle 802 can have a predetermined rotational orientationdefined by a first rotational angle between a lateral plane 884 parallelto the lateral axis 781 and a dimension of the shaped abrasive particle802. Notably, reference herein to a dimension of the shaped abrasiveparticle 802 can include reference to a bisecting axis 831 of the shapedabrasive particle 802, such bisecting axis 831 extending through acenter point 821 of the shaped abrasive particle 802 along a surface(e.g., a side or an edge) connected to (directly or indirectly) thebacking 801. Accordingly, in the context of a shaped abrasive particlepositioned in a side orientation, (see, e.g., FIG. 6), the bisectingaxis 831 can extend through a center point 821 and in the direction ofthe width (w) of a side 833 closest to the surface of the backing 801.

In certain embodiments, the predetermined rotational orientation of theshaped abrasive particle 802 can be defined by a predeterminedrotational angle 841 that defines the smallest angle between thebisecting axis 831 and the lateral plane 884, both of which extendthrough the center point 821 as viewed from the top down in FIG. 8A. Inaccordance with an embodiment, the predetermined rotational angle 841,and thus the predetermined rotational orientation, can be 0°. In otherembodiments, the predetermined rotational angle defining thepredetermined rotational orientation can be greater, such as at leastabout 2°, at least about 5°, at least about 10°, at least about 15°, atleast about 20°, at least about 25°, at least about 30°, at least about35°, at least about 40°, at least about 45°, at least about 50°, atleast about 55°, at least about 60°, at least about 70°, at least about80°, or even at least about 85°. Still, the predetermined rotationalorientation as defined by the rotational angle 841 may be not greaterthan about 90°, such as not greater than about 85°, not greater thanabout 80°, not greater than about 75°, not greater than about 70°, notgreater than about 65°, not greater than about 60°, such as not greaterthan about 55°, not greater than about 50°, not greater than about 45°,not greater than about 40°, not greater than about 35°, not greater thanabout 30°, not greater than about 25°, not greater than about 20°, suchas not greater than about 15°, not greater than about 10°, or even notgreater than about 5°. It will be appreciated that the predeterminedrotational orientation can be within a range between any of the aboveminimum and maximum angles.

FIG. 8B includes a perspective view illustration of a portion of theabrasive article 800 including the shaped abrasive particle 802 having atriangular two-dimensional shape. The referenced shaped abrasiveparticle having a triangular two-dimensional shape is merelyillustrative, and it will be appreciated that any shaped abrasiveparticle having any of the shapes of the embodiments herein can besubstituted for the triangular shaped abrasive particle of FIG. 8B. Asillustrated, the abrasive article 800 can include the shaped abrasiveparticle 802 overlying the backing 801 in a first position 812 such thatthe shaped abrasive particle 802 includes a first rotational orientationrelative to the lateral axis 781 defining the width of the backing 801.Certain aspects of the predetermined orientation of a shaped abrasiveparticle may be described by reference to a x, y, z three-dimensionalaxis as illustrated. For example, the predetermined longitudinalorientation of the shaped abrasive particle 802 may be described byreference to the position of the shaped abrasive particle 802 relativeto the y-axis, which extends parallel to the longitudinal axis 780 ofthe backing 801. Moreover, the predetermined lateral orientation of theshaped abrasive particle 802 may be described by reference to theposition of the shaped abrasive particle on the x-axis, which extendsparallel to the lateral axis 781 of the backing 801. Furthermore, thepredetermined rotational orientation of the shaped abrasive particle 802may be defined with reference to a bisecting axis 831 that extendsthrough the center point 821 of the side 833 of the shaped abrasiveparticle 802. Notably, the side 833 of the shaped abrasive particle 802may be connected either directly or indirectly to the backing 801. In aparticular embodiment, the bisecting axis 831 may form an angle with anysuitable reference axis including, for example, the x-axis that extendsparallel to the lateral axis 781. The predetermined rotationalorientation of the shaped abrasive particle 802 may be described as arotational angle formed between the x-axis and the bisecting axis 831,which rotational angle is depicted in FIG. 8B as angle 841. Notably, thecontrolled placement of a plurality of shaped abrasive particles on thebacking of the abrasive article may facilitate improved performance ofthe abrasive article.

FIG. 9 includes a perspective view illustration of a portion of anabrasive article including shaped abrasive particles havingpredetermined orientation characteristics relative to a grindingdirection in accordance with an embodiment. Notably, as with FIG. 8B,the shaped abrasive particles have a triangular two-dimensional shape,which is done merely for illustration and discussion of certain featuresof the abrasive article. It will be appreciated that any of shapedabrasive particles of the embodiments herein can be substituted for theshaped abrasive particles illustrated in FIG. 9. In one embodiment, theabrasive article 900 can include a shaped abrasive particle 902 having apredetermined orientation relative to another shaped abrasive particle903 and/or relative to a grinding direction 985. The grinding direction985 may be an intended direction of movement of the abrasive articlerelative to a workpiece in a material removal operation. In particularinstances, the grinding direction 985 may be defined relative to thedimensions of the backing 901. For example, in one embodiment, thegrinding direction 985 may be substantially perpendicular to the lateralaxis 981 of the backing and substantially parallel to the longitudinalaxis 980 of the backing 901. The predetermined orientationcharacteristics of the shaped abrasive particle 902 may define aninitial contact surface of the shaped abrasive particle 902 with aworkpiece. For example, the shaped abrasive particle 902 can includemajor surfaces 963 and 964 and side surfaces 965 and 966, each of whichcan extend between the major surfaces 963 and 964. The predeterminedorientation characteristics of the shaped abrasive particle 902 canposition the particle 902 such that the major surface 963 is configuredto make initial contact with a workpiece before the other surfaces ofthe shaped abrasive particle 902 during a material removal operation.Such an orientation may be considered a major surface orientationrelative to the grinding direction 985. More particularly, the shapedabrasive particle 902 can have a bisecting axis 931 having a particularorientation relative to the grinding direction 985. For example, asillustrated, the vector of the grinding direction 985 and the bisectingaxis 931 are substantially perpendicular to each other. It will beappreciated that, just as any range of predetermined rotationalorientations relative to the backing are contemplated for a shapedabrasive particle, any range of orientations of the shaped abrasiveparticles relative to the grinding direction 985 are contemplated andcan be utilized.

The shaped abrasive particle 903 can have one or more differentpredetermined orientation characteristics as compared to the shapedabrasive particle 902 and the grinding direction 985. As illustrated,the shaped abrasive particle 903 can include major surfaces 991 and 992,each of which can be joined by side surfaces 971 and 972. Moreover, asillustrated, the shaped abrasive particle 903 can have a bisecting axis973 forming a particular angle relative to the vector of the grindingdirection 985. As illustrated, the bisecting axis 973 of the shapedabrasive particle 903 can have a substantially parallel orientation withthe grinding direction 985 such that the angle between the bisectingaxis 973 and the grinding direction 985 is essentially 0 degrees.Accordingly, the predetermined orientation characteristics of the shapedabrasive particle 903 facilitate initial contact of the side surface 972with a workpiece before any of the other surfaces of the shaped abrasiveparticle 903. Such an orientation of the shaped abrasive particle 903may be considered a side surface orientation relative to the grindingdirection 985.

Still, in one non-limiting embodiment, it will be appreciated that anabrasive article can include one or more groups of shaped abrasiveparticles that can be arranged in one or more predetermineddistributions relative to the backing, a grinding direction, and/or eachother. For example, one or more groups of shaped abrasive particles, asdescribed herein, can have a predetermined orientation relative to agrinding direction. Moreover, the abrasive articles herein can have oneor more groups of shaped abrasive particles, each of the groups having adifferent predetermined orientation relative to a grinding direction.Utilization of groups of shaped abrasive particles having differentpredetermined orientations relative to a grinding direction mayfacilitate improved performance of the abrasive article.

FIG. 10 includes a top view illustration of a portion of an abrasivearticle in accordance with an embodiment. In particular, the abrasivearticle 1000 can include a first group 1001 including a plurality ofshaped abrasive particles. As illustrated, the shaped abrasive particlescan be arranged relative to each other one the backing 101 to define apredetermined distribution. More particularly, the predetermineddistribution can be in the form of a pattern 1023 as viewed top-down,and more particularly defining a triangular shaped two-dimensionalarray. As further illustrated, the first group 1001 can be arranged onthe abrasive article 1000 defining a predetermined macro-shape 1031overlying the backing 101. In accordance with an embodiment, themacro-shape 1031 can have a particular two-dimensional shape as viewedtop-down. Some exemplary two-dimensional shapes can include polygons,ellipsoids, numerals, Greek alphabet characters, Latin alphabetcharacters, Russian alphabet characters, Arabic alphabet characters,Kanji characters, complex shapes, irregular shapes, designs, any acombination thereof. In particular instances, the formation of a grouphaving a particular macro-shape may facilitate improved performance ofthe abrasive article.

As further illustrated, the abrasive article 1000 can include a group1004 including a plurality of shaped abrasive particles which can bearranged on the surface of the backing 101 relative to each other todefine a predetermined distribution. Notably, the predetermineddistribution can include an arrangement of the plurality of the shapedabrasive particles that define a pattern 422, and more particularly, agenerally quadrilateral pattern. As illustrated, the group 1004 candefine a macro-shape 1034 on the surface of the abrasive article 1000.In one embodiment, the macro-shape 1034 of the group 1004 can have atwo-dimensional shape as viewed top down, including for example apolygonal shape, and more particularly, a generally quadrilateral(diamond) shape as viewed top down on the surface of the abrasivearticle 1000. In the illustrated embodiment of FIG. 10, the group 1001can have a macro-shape 1031 that is substantially the same as themacro-shape 1034 of the group 1004. However, it will be appreciated thatin other embodiments, various different groups can be used on thesurface of the abrasive article, and more particularly wherein each ofthe different groups has a different macro-shape relative to each other.

As further illustrated, the abrasive article can include groups 1001,1002, 1003, and 1004 which can be separated by channel regions 1021 and1024 extending between the groups 1001-1004. In particular instances,the channel regions 1021 and 1024 can be substantially free of shapedabrasive particles. Moreover, the channel regions 1021 and 1024 may beconfigured to move liquid between the groups 1001-1004 and furtherimprove swarf removal and grinding performance of the abrasive article.Furthermore, in a certain embodiment, the abrasive article 1000 caninclude channel regions 1021 and 1024 extending between groups1001-1004, wherein the channel regions 1021 and 1024 can be patterned onthe surface of the abrasive article 1000. In particular instances, thechannel regions 1021 and 1024 can represent a regular and repeatingarray of features extending along a surface of the abrasive article.

The fixed abrasive articles of the embodiments herein can be utilized invarious material removal operations. For example, fixed abrasivearticles herein can be used in methods of removing material from aworkpiece by moving the fixed abrasive article relative to theworkpiece. The relative movement between the fixed abrasive and theworkpiece can facilitate removal of the material from the surface of theworkpiece. Various workpieces can be modified using the fixed abrasivearticles of the embodiments herein, including but not limited to,workpieces comprising inorganic materials, organic materials, and acombination thereof. In a particular embodiment, the workpiece mayinclude a metal, such as a metal alloy. In one particular instance, theworkpiece can consist essentially of a metal or metal alloy, such asstainless steel.

According to another embodiment, the shaped abrasive particles of theembodiments herein can be incorporated into a fixed abrasive article,which may facilitate improved performance of the fixed abrasive article.In at least one embodiment, a coated abrasive article can include theshaped abrasive particles of the embodiments herein and can have aparticularly improved performance. FIG. 19 includes a generalized plotof specific grinding energy versus cumulative material removed for threecoated abrasive articles. In particular, FIG. 19 includes a plot 1901defining an initial specific grinding energy at the intersection of theplot 1901 with the Y-axis (i.e., specific grinding energy). The plot1901 further has a full life value, defined by the greatest value ofcumulative material removed on the plot 1901 (i.e., the cumulativematerial removed value at the end of the plot 1901). The plot 1901further defines a half-life point 1902 defining the point on the curvethat is half of the full life value. The half-life point 1902 can have acorresponding specific grinding energy value given by the intersectionof the dotted line at the Y-axis.

FIG. 19 further includes a plot 1920 having an initial specific grindingenergy at the intersection of the plot 1920 with the Y-axis (i.e.,specific grinding energy). The plot 1920 further has a full life value,defined by the greatest value of cumulative material removed on the plot1920 (i.e., the cumulative material removed value at the end of the plot1920). The plot 1920 further defines a half-life point 1921 defining thepoint on the curve that is half of the full life value. The half-lifepoint 1921 can have a corresponding specific grinding energy value givenby the intersection of the dotted line at the Y-axis. Moreover, the plot1920 can have a minimum point 1922 defined by the lowest value ofspecific grinding energy on the plot 1920. For certain plots, such asplot 1901, the minimum point is the same as the initial grinding energyat the intersection of the plot 1901 with the Y-axis.

FIG. 19 further includes a plot 1930 having an initial specific grindingenergy at the intersection of the plot 1930 with the Y-axis (i.e.,specific grinding energy). The plot 1930 further has a full life value,defined by the greatest value of cumulative material removed on the plot1930 (i.e., the cumulative material removed value at the end of the plot1930). The plot 1930 further defines a half-life point 1931 defining thepoint on the curve that is half of the full life value. The half-lifepoint 1931 can have a corresponding specific grinding energy value givenby the intersection of the dotted line at the Y-axis. Moreover, the plot1930 can have a minimum point 1932 defined by the lowest value ofspecific grinding energy on the plot 1930. Unlike the plots 1901 and1920, the plot 1930 demonstrates a significant decrease in the specificgrinding energy in the initial stages of grinding. This may facilitatemore efficient grinding in the initial stages defined by a lower minimumpoint 1932 and half-life point 1931 compared to those of the plots 1901and 1920.

According to one embodiment, a fixed abrasive article can includeabrasive particles, such as the shaped abrasive particles of theembodiments herein, wherein the fixed abrasive article can have ahalf-life/initial energy factor of not greater than 1. Thehalf-life/initial energy factor is calculated by dividing the specificgrinding energy at the half-life point of the plot by the initialspecific grinding energy of the fixed abrasive (i.e., the point at whichthe plot intersects the Y-axis). For example, for the plot 1901 of FIG.19, the half-life/initial energy factor would be greater than 1, sincethe specific grinding energy at the half-life point 1902 is greater thanthe initial specific grinding energy. Moreover, for plot 1920, thehalf-life/initial energy factor would be greater than 1 because thespecific grinding energy at the half-life point 1921 is greater than theinitial specific grinding energy of the plot 1920. However, thehalf-life/initial energy factor for plot 1930 would be less than 1, asthe specific grinding energy at the half-life point 1931 issignificantly less than the initial specific grinding energy of the plot1930.

In more particular terms, the embodiments herein may include a fixedabrasive article having a half-life/initial energy factor of not greaterthan 1.2, such as not greater than 1.18 or not greater than 1.16 or notgreater than 1.14 or not greater than 1.12 or not greater than 1.10 ornot greater than 1.08 or not greater than 1.06 or not greater than 1.05or not greater than 1.04 or not greater than 1.03 or not greater than1.02 or not greater than 1.01 or not greater than 1.00, or not greaterthan 0.99 or not greater than 0.98 or not greater than 0.97 or notgreater than 0.96 or not greater than 0.95 or not greater than 0.94 ornot greater than 0.93 or not greater than 0.92 or not greater than 0.91or not greater than 0.9 or not greater than 0.89 or not greater than0.88 or not greater than 0.87 or not greater than 0.86 or not greaterthan 0.85 or not greater than 0.84 or not greater than 0.83 or notgreater than 0.82 or not greater than 0.81 or not greater than 0.8 ornot greater than 0.79 or not greater than 0.78 or not greater than 0.77or not greater than 0.76 or not greater than 0.75 or not greater than0.74 or not greater than 0.73 or not greater than 0.72 or not greaterthan 0.71 or not greater than 0.7 or not greater than 0.69 or notgreater than 0.68 or not greater than 0.67 or not greater than 0.66 ornot greater than 0.65 or not greater than 0.64 or not greater than 0.63or not greater than 0.62 or not greater than 0.61 or not greater than0.6 or not greater than 0.55 or even not greater than 0.5. Still, inanother embodiment, the fixed abrasive articles herein (e.g., a coatedabrasive article) can have a half-life/initial energy factor of at least0.01 or at least 0.1 or at least 0.15 or at least 0.2 or at least 0.25or at least 0.3 or at least 0.35 or at least 0.4 or at least 0.45 or atleast 0.5 or at least 0.55 or at least 0.6 or at least 0.65 or at least0.7 or at least 0.75 or at least 0.8 or at least 0.85 or at least 0.9.It will be appreciated that the half-life/initial energy factor can bewithin a range including any of the minimum and maximum values notedabove. Moreover, the method of analyzing the half-life/initial energyfactor can be conducted according to a standardized material removaltest as provided in the Examples herein.

In yet another embodiment, a fixed abrasive article can include one ormore shaped abrasive particle of the embodiments herein and have aparticular minimum specific grinding energy according to thestandardized material removal test. For example, the fixed abrasive canhave a minimum specific grinding energy factor of at least 5%, whereinthe minimum specific grinding energy factor is calculated by theequation [(Ei−Em)/Ei]×100%, wherein Ei represents the initial specificgrinding energy of the fixed abrasive as it intersects the Y-axis, andEm represents the specific grinding energy at the minimum point on theplot. According to one embodiment, the fixed abrasive can have a minimumspecific grinding energy factor of at least 5.5% or at least 6% or atleast 6.5% or at least 7% or at least 7.5% or at least 8% or at least8.5% or at least 9% or at least 9.5% or at least 10% or at least 10.5%or at least 11% or at least 11.5% or at least 12% or at least 12.5% orat least 13% or at least 13.5% or at least 14% or at least 14.5%. Still,in another embodiment, the minimum specific grinding energy factor canbe not greater than 60% or not greater than 50% or not greater than 40%or not greater than 30% or not greater than 25% or not greater than 20%or not greater than 18% or not greater than 15%. It will be appreciatedthat the minimum specific grinding energy factor can be within a rangebetween any of the minimum and maximum percentages noted above. It willbe appreciated that those samples having a specific grinding energy atthe minimum point that is the same as the initial grinding energy (e.g.,plot 1901) the value of the minimum specific grinding energy factor is0%.

The fixed abrasive articles of the embodiments herein, having theparticular features of minimum specific grinding energy factor andhalf-life/initial energy factor can include any one or combination offeatures described in the other embodiments herein.

EXAMPLES Example 1

Six samples of shaped abrasive particles were created and tested forcomparison of performance. A first sample, Sample S1, was initiallyformed from a mixture including approximately 45-50 wt % boehmite. Theboehmite was obtained from Sasol Corp. as Catapal B and modified byautoclaving a 30% by weight mixture of the Catapal B with deionizedwater and nitric acid. The nitric acid-to-boehmite ratio wasapproximately 0.025 in the autoclave and treated at 100° C. to 250° C.for a time ranging from 5 minutes to 24 hours. The autoclaved Catapal Bsol was then dried by conventional means. One may also use analternative boehmite, commercially available as Disperal from SasolCorp. The boehmite was mixed and seeded with 1% alpha alumina seedsrelative to the total alumina content of the mixture. The alpha aluminaseeds were made by milling of corundum using conventional techniques,described for example in U.S. Pat. No. 4,623,364. The mixture alsoincluded 45-50 wt % water and 2.5-7 wt % additional nitric aciddepending on the desired viscosity of the mixture, which were used toform the gel mixture. The ingredients were mixed in a planetary mixer ofconventional design and mixed under reduced pressure to remove gaseouselements from the mixture (e.g., bubbles).

After gelling, the mixture was deposited by hand into openings of aproduction tool made of stainless steel. The openings in the productiontool were open to both sides of the production tool, such that they wereapertures extending through the entire thickness of the production tool.The cavities or openings of the production tool had a shapeapproximately the same as the shape of the particles provided herein.All samples were made with a production tool made of stainless steel.The surfaces of the openings in the production tool were coated with alubricant of olive oil to facilitate removal of the precursor shapedabrasive particles from the production tool. The gel was placed in theopenings of the screen and dried at room temperature for at least 12hours. After drying, the precursor shaped abrasive particles wereremoved from the screen and sintered between 1250-1400° C. forapproximately 10 minutes.

The shaped abrasive particles of Sample S1 had a two-dimensional shapeof an equilateral triangle as provided in the image of FIG. 20, havingan average width of 1400 microns and a height of approximately 300microns. The body was formed essentially of a seeded sol-gel aluminamaterial having an average grain size of less than 1 micron. The shapedabrasive particles of Sample S1 had an average strength of approximately847 MPa, an average tip sharpness of approximately 20 microns, a ShapeIndex of approximately 0.5, and a 3SF of approximately 1.7.

A second sample, Sample S2, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S2 includedshaped abrasive particles having a two-dimensional shape as provided inthe image of FIG. 21, which includes a body having a partially-concaveshape according to the embodiments herein. The body had an average widthof 1500 microns and a height of approximately 300 microns. The body wasformed essentially of a seeded sol-gel alumina material having anaverage grain size of less than 1 micron. The shaped abrasive particlesof Sample S2 had an average strength of approximately 847 MPa, anaverage tip sharpness of approximately 20 microns, a Shape Index ofapproximately 0.35, and a 3SF of approximately 0.8.

A third sample, Sample S3, which may represent a conventional shapedabrasive particle, was formed using the same process used to form theshaped abrasive particles of Sample S1. Sample S3 includes shapedabrasive particles having a two-dimensional shape as provided in theimage of FIG. 22, which is a completely concave triangular shape. Thebody was formed essentially of a seeded sol-gel alumina material havingan average grain size of less than 1 micron. The body had an averagewidth of 1500 microns and a height of approximately 300 microns. Theshaped abrasive particles of Sample S3 had an average strength ofapproximately 847 MPa, an average tip sharpness of 20 microns, a ShapeIndex of approximately 0.38, and a 3SF of approximately 1.0.

A fourth sample, Sample CS4, was a conventional shaped abrasive particlecommercially available as 3M984F from 3M Corporation. The body had anaverage width of 1400 microns and a height of approximately 300 microns.The shaped abrasive particles of Sample CS4 had a rare-earth elementdoped alpha-alumina composition, an average tip sharpness ofapproximately 20 microns, an average strength of approximately 606 MPa,a Shape Index of 0.5, and a 3SF of approximately 1.2. FIG. 23 includesan image of a shaped abrasive particle from Sample CS4.

A fifth sample, Sample S5, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S5 includedshaped abrasive particles having a two-dimensional shape as provided inthe image of FIG. 24, which includes a body having a partially-concaveshape according to the embodiments herein. The body had an average widthof 1500 microns and a height of approximately 330 microns. The body wasformed essentially of a seeded sol-gel alumina material having anaverage grain size of less than 1 micron. The shaped abrasive particlesof Sample S5 had an average strength of approximately 847 MPa, anaverage tip sharpness of approximately 20 microns, a Shape Index ofapproximately 0.43, and a 3SF of approximately 1.25.

A sixth sample, Sample S6 included shaped abrasive particles having agenerally arrowhead shape as illustrated in FIG. 25. The shaped abrasiveparticles of Sample S6 were sourced from a group of shaped abrasiveparticles having a generally triangular shape and represent grains wherethe opening of the production tool was not completely filled, thusproducing the shaped abrasive particles illustrated. The particles weregenerally formed from the gel as provided in Sample S1, but were notformed by hand. The body had an average width of 1500 microns and aheight of approximately 330 microns. The body was formed essentially ofa seeded sol-gel alumina material having an average grain size of lessthan 1 micron. The shaped abrasive particles of Sample S6 had an averagestrength of approximately 847 MPa estimated from the microstructure, anaverage tip sharpness of approximately 20 microns, a Shape Index ofapproximately 0.42, and a 3SF of approximately 1.2. Note that Sample S6was tested in two different side orientations. In the first sideorientation (“Side-C”), the curved, oblique side section of the sidesurface defined the leading edge of the corner during the grinding test.In the second side orientation (“Side-S”), a linear surface section ofthe side surface was the leading edge of the external corner during thegrinding test.

All samples were tested according to a single grit grinding test (SGGT)in a major surface orientation and side orientation. In conducting theSGGT, one single shaped abrasive particle is held in a grit holder by abonding material of epoxy. The shaped abrasive particle is secured inthe desired orientation (i.e., major surface orientation or side surfaceorientation) and moved across a workpiece of 304 stainless steel for ascratch length of 8 inches using a wheel speed of 22 m/s and an initialscratch depth of 30 microns. The shaped abrasive particle produces agroove in the workpiece having a cross-sectional area (AR). For eachsample set, each shaped abrasive particle completes 15 passes across the8 inch length, 10 individual particles are tested for each of theorientation and the results are analyzed. The test measures thetangential force exerted by the grit on the workpiece, in the directionthat is parallel to the surface of the workpiece and the direction ofthe groove, and the net change in the cross-sectional area of the groovefrom beginning to the end of the scratch length is measured to determinethe shaped abrasive particle wear. The net change in the cross-sectionalarea of the groove for each pass can be measured. For the SGGT, the netcross-sectional area of the groove is defined as the difference betweenthe cross-sectional area of the groove below the surface and the crosssectional area of the material displaced above the surface. Performance(Ft/A) is defined as the ratio of the tangential force to the netcross-sectional area of the groove.

The SGGT is conducted using two different orientations of the shapedabrasive particles relative to the workpiece. The SGGT is conducted witha first sample set of shaped abrasive particles in a major surfaceorientation (i.e., “front” in FIG. 9), wherein a major surface of eachshaped abrasive particle is oriented perpendicular to the grindingdirection such that the major surface initiates grinding on theworkpiece. The results of the SGGT using the sample set of shapedabrasive particles in a major surface orientation allows for measurementof the grinding efficiency of the shaped abrasive particles in a majorsurface orientation.

The SGGT is also conducted with a second sample set of shaped abrasiveparticles in a side surface orientation (i.e., “side” in FIG. 9),wherein a side surface of each shaped abrasive particle is orientedperpendicular to the grinding direction such that the side surfaceinitiates grinding of the workpiece. The results of the SGGT test usingthe sample set of shaped abrasive particles in a side orientation allowsfor measurement of the grinding efficiency of the shaped abrasiveparticles in a side orientation.

FIG. 26 includes a plot of median force per total area removed from theworkpiece, which is representative of data derived from the SGGT for allof the samples. The median force per total area is averaged for thefront (i.e., major surface orientation) and side (i.e., side surfaceorientation) orientations. The force per total area removed is a measureof the grinding efficiency of the shaped abrasive particles, with alower force per total area removed indicating more efficient grindingperformance. As illustrated, Sample S2 demonstrated the best performanceof all samples tested. Without wishing to be tied to a particular theoryit is noted that the combination of strength, tip sharpness, and ShapeIndex of the shaped abrasive particles of Sample S2 is superior over allother samples. Unexpectedly and quite remarkably, Sample S2 demonstrateda 15% improvement in grinding efficiency over the shaped abrasiveparticles of Sample CS4, a 40% improvement in grinding efficiencycompared to the shaped abrasive particles of Sample S3, and nearly a 60%improvement compared to the shaped abrasive particles of Sample S1.

Example 2

Abrasive particles having the shape and microstructure as outlined inSamples S1 and S2 were formed using a machine including a die to extrudethe gel mixture into openings of a production tool being translatedunder the die. These grains were used to form coated abrasive sampleshaving the construction outlined below and designated CAS1 and CAS2,respectively. A comparative belt commercially available from 3M as CII984F and designated CACS4. The samples CAS1 and CAS2 had the sameconstruction, which is provided below. A backing of finished cloth of 47pounds per ream was obtained and coated with a make formulationincluding a phenol formaldehyde resin as provided in Table 4. Using anelectrostatic deposition process, 41 pounds per ream of abrasiveparticles having substantially the same shape and microstructure asoutlined for Samples S1 or S2 were applied to the backing with the makecoat. The structure was dried in an oven for two hours at 80° C. It willbe appreciated that the make coat was created such that sum of thecomponents provided in Table 4 equals 100%.

TABLE 4 Make Coat Formulation Make Formulation Component PercentageFiller NYAD Wollastonite 45-50 wt % 400 Wet Witcona 1260 0.10-.2 wt %Resin, SI 45-50 wt % Solmod Silane A1100 0.1-3 wt % Water 0.1-1 wt %

The coated abrasive structures were then coated with a size coat havingthe formulation presented in Table 5. The construction was heat treatedin an oven set for a final soak temperature of 100-120° C., in which thesample was held for approximately 20-30 minutes. It will be appreciatedthat the size coat was created such that sum of the components providedin Table 5 equals 100%.

TABLE 5 Size Coat Formulation Size Formulation Component Percentage Dye2-4 wt % Solmod Tamol 165A 0.5-2 wt % Filler Syn Cryolite K 40-45 wt %Resin Single Comp 94-908 50-55 wt % DF70 Defoamer 0.1-0.2 wt % Water 2-4wt %

The coated abrasive sample was then placed into an oven to undergo heattreatment. The oven temperature was set for a final soak temperature ofapproximately 110-120° C., in which the sample was held forapproximately 10-12 hours.

A supersize coat having the formulation provided below in Table 6 wasthen applied to the Samples CAS1 and CAS2 and processed in the samemanner as the size coat. It will be appreciated that the supersize coatwas created such that sum of the components provided in Table 6 equals100%.

TABLE 6 Supersize Coat Formulation Supersize Formulation ComponentPercentage Dye 1-3 wt % Solmod Cabosil 0.05-3 wt % Solmod DAXAD 11 1-4wt % Filler Type A 63-67 wt % Resin PF Prefere 80-5080A 20-25 wt % DF70Defoamer 0.1-0.2 wt % Water 6-10 wt %

Each of the three different coated abrasive samples CACS4, CAS1, andCAS2 was tested according to a standardized grinding test using theconditions summarized in Table 7. Notably, two sample coated abrasiveswere tested in each case to derive the results.

TABLE 7 Test mode: Dry, straight plunge Test conditions: Constant MRR′ =4 inch³/min/inch Belt speed (Vs) = 7500 sfpm (38 m/s) Work material: 304ss Hardness: 96-104 HRB Size: 0.5 × 0.5 × 12 inches Contact width: 0.5in Contact Wheel: Steel Measurements: Power, Grinding Forces, MRR′ andSGE Cum MR compared at SGE = 2.4 Hp.min/inch³

FIG. 27 includes a plot of specific grinding energy versus cumulativematerial removed (at a material removal rate of 4 inch³/min inch) foreach of the samples. It is notable and quite remarkable that the coatedabrasive utilizing the abrasive particles of Sample S2 had asignificantly lower specific grinding energy for the initial phase ofthe grinding test. CAS2 demonstrated a significantly lower specificgrinding energy for a significant portion of the life of the abrasivearticle compared to CACS4 and CAS1. In particular, CAS2 demonstrated aminimum specific grinding energy factor of approximately 6% and ahalf-life initial energy factor of approximately 1.05. Sample CACS4demonstrated a minimum specific grinding energy factor of 0% and ahalf-life initial energy factor of approximately 1.1. Sample CAS1demonstrated a minimum specific grinding energy factor of 0% and ahalf-life initial energy factor of approximately 1.2.

The present application represents a departure from the state of theart. Conventional shaped abrasive particles have previously focused onmaking triangular shaped grains having the sharpest possible corners andedges. However, through empirical studies of shaped abrasive particleshaving various shapes and microstructures, it has been discovered thatcertain grain features (e.g., tip sharpness, strength, and Shape Index)appear to be interrelated and may be controlled with respect to eachother to provide improved performance of a shaped abrasive particle.Additionally, as noted herein, the height may be related as well.Notably, in the present application, it is noted that one may notnecessarily need to create a shaped abrasive particle with the sharpestfeatures, but instead may control one or more of a combination of grainfeatures, including tip sharpness, strength, Shape Index, and heightrelative to each other to improve the grinding performance of a shapedabrasive particle beyond conventional shaped abrasive particles. Inparticular, it is noted that the Shape Index may define an overall shapeof the body and how stress is distributed throughout the body duringgrinding, which when combined with a suitable tip sharpness andstrength, may provide improved results over conventional triangularshaped abrasive particles having sharp tips. Moreover, while notcompletely understood and without wishing to be tied to a particulartheory, it is thought that one or a combination of these features of theembodiments described herein facilitate the remarkable and unexpectedperformance of these particles in fixed abrasives, such as coatedabrasive and bonded abrasives.

Certain features, for clarity, described herein in the context ofseparate embodiments, may also be provided in combination in a singleembodiment. Conversely, various features that are, for brevity,described in the context of a single embodiment, may also be providedseparately or in any subcombination. Further, reference to values statedin ranges includes each and every value within that range.

Benefits, other advantages, and solutions to problems have beendescribed above with regard to specific embodiments. However, thebenefits, advantages, solutions to problems, and any feature(s) that maycause any benefit, advantage, or solution to occur or become morepronounced are not to be construed as a critical, required, or essentialfeature of any or all the claims.

The specification and illustrations of the embodiments described hereinare intended to provide a general understanding of the structure of thevarious embodiments. The specification and illustrations are notintended to serve as an exhaustive and comprehensive description of allof the elements and features of apparatus and systems that use thestructures or methods described herein. Separate embodiments may also beprovided in combination in a single embodiment, and conversely, variousfeatures that are, for brevity, described in the context of a singleembodiment, may also be provided separately or in any subcombination.Further, reference to values stated in ranges includes each and everyvalue within that range. Many other embodiments may be apparent toskilled artisans only after reading this specification. Otherembodiments may be used and derived from the disclosure, such that astructural substitution, logical substitution, or another change may bemade without departing from the scope of the disclosure. Accordingly,the disclosure is to be regarded as illustrative rather thanrestrictive.

The description in combination with the figures is provided to assist inunderstanding the teachings disclosed herein. The following discussionwill focus on specific implementations and embodiments of the teachings.This focus is provided to assist in describing the teachings and shouldnot be interpreted as a limitation on the scope or applicability of theteachings. However, other teachings can certainly be used in thisapplication.

As used herein, the terms “comprises,” “comprising,” “includes,”“including,” “has,” “having” or any other variation thereof, areintended to cover a non-exclusive inclusion. For example, a method,article, or apparatus that comprises a list of features is notnecessarily limited only to those features but may include otherfeatures not expressly listed or inherent to such method, article, orapparatus. Further, unless expressly stated to the contrary, “or” refersto an inclusive-or and not to an exclusive-or. For example, a conditionA or B is satisfied by any one of the following: A is true (or present)and B is false (or not present), A is false (or not present) and B istrue (or present), and both A and B are true (or present).

Also, the use of “a” or “an” is employed to describe elements andcomponents described herein. This is done merely for convenience and togive a general sense of the scope of the invention. This descriptionshould be read to include one or at least one and the singular alsoincludes the plural, or vice versa, unless it is clear that it is meantotherwise.

For example, when a single item is described herein, more than one itemmay be used in place of a single item. Similarly, where more than oneitem is described herein, a single item may be substituted for that morethan one item.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. The materials, methods, andexamples are illustrative only and not intended to be limiting. To theextent not described herein, many details regarding specific materialsand processing acts are conventional and may be found in reference booksand other sources within the structural arts and correspondingmanufacturing arts.

The above-disclosed subject matter is to be considered illustrative, andnot restrictive, and the appended claims are intended to cover all suchmodifications, enhancements, and other embodiments, which fall withinthe true scope of the present invention. Thus, to the maximum extentallowed by law, the scope of the present invention is to be determinedby the broadest permissible interpretation of the following claims andtheir equivalents, and shall not be restricted or limited by theforegoing detailed description.

The Abstract of the Disclosure is provided to comply with Patent Law andis submitted with the understanding that it will not be used tointerpret or limit the scope or meaning of the claims. In addition, inthe foregoing Detailed Description of the Drawings, various features maybe grouped together or described in a single embodiment for the purposeof streamlining the disclosure. This disclosure is not to be interpretedas reflecting an intention that the claimed embodiments require morefeatures than are expressly recited in each claim. Rather, as thefollowing claims reflect, inventive subject matter may be directed toless than all features of any of the disclosed embodiments. Thus, thefollowing claims are incorporated into the Detailed Description of theDrawings, with each claim standing on its own as defining separatelyclaimed subject matter.

Items:

Item 1. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, whereinthe body comprises a sharpness-shape-strength factor (3SF) within arange between about 0.7 and about 1.7 and a Shape Index within a rangebetween at least about 0.01 and not greater than about 0.47.

Item 2. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, whereinthe body comprises a Shape Index within a range between at least about0.01 and not greater than about 0.47 and a strength within a rangebetween at least about 350 MPa and not greater than about 1500 MPa.

Item 3. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, whereinthe body comprises an average tip sharpness within a range between notgreater than about 80 microns and at least about 1 micron, a Shape Indexwithin a range between at least about 0.01 and not greater than about0.47, and the body comprises a strength of at least about 350 MPa andnot greater than about 1500 MPa.

Item 4. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, wherein afirst portion of the side surface has a partially-concave shape.

Item 5. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, wherein afirst portion of the side surface extends between a first corner andsecond corner of the body that are adjacent to each other, and whereinthe first portion of the side surface comprises a first curved sectionjoined to a first linear section.

Item 6. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, wherein afirst portion of the side surface comprises a first curved sectionjoined to a first linear section and defining an interior cornerdefining an obtuse angle.

Item 7. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, whereinthe body is a hybrid polygonal shape having a sum of the externalcorners of substantially 180 degrees, and further comprising a firstportion of the side surface having a first curved section.

Item 8. A shaped abrasive particle comprising a body comprising a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, whereinthe body comprises a first maximum tip width (Wt1) that defines amaximum width of a first arm of the body, and wherein the first maximumtip width (Wt1) is disposed a distance from a first terminal end of thefirst arm and between a midpoint of the body and the first terminal end.

Item 9. The shaped abrasive particle of any one of items 4, 5, 6, 7, and8, wherein the body comprises a Shape Index within a range between atleast about 0.01 and not greater than about 0.47.

Item 10. The shaped abrasive particle of any one of items 1, 2, 3, and9, wherein the body comprises a Shape Index of at least about 0.02, atleast about 0.03, at least about 0.04, at least about 0.05, at leastabout 0.06, at least about 0.07, at least about 0.08, at least about0.09, at least about 0.10, at least about 0.11, at least about 0.12, atleast about 0.13, at least about 0.14, at least about 0.15, at leastabout 0.16, at least about 0.17, at least about 0.18, at least about0.19, at least about 0.20, at least about 0.21, at least about 0.22, atleast about 0.23, at least about 0.24, at least about 0.25, at leastabout 0.26, at least about 0.27, at least about 0.28, at least about0.29, at least about 0.30, at least about 0.31, at least about 0.32, atleast about 0.33, at least about 0.34, at least about 0.35, at leastabout 0.36, at least about 0.37, at least about 0.38, at least about0.39, at least about 0.40, at least about 0.41, at least about 0.42, atleast about 0.43, at least about 0.44, at least about 0.45, at leastabout 0.46.

Item 11. The shaped abrasive particle of any one of items 1, 2, 3, and9, wherein the body comprises a Shape Index of not greater than about0.46, not greater than about 0.45, not greater than about 0.44, notgreater than about 0.43, not greater than about 0.42, not greater thanabout 0.41, not greater than about 0.40, not greater than about 0.39,not greater than about 0.38, not greater than about 0.37, not greaterthan about 0.36, not greater than about 0.35, not greater than about0.34, not greater than about 0.33, not greater than about 0.32, notgreater than about 0.31, not greater than about 0.30, not greater thanabout 0.29, not greater than about 0.28, not greater than about 0.27,not greater than about 0.26, not greater than about 0.25, not greaterthan about 0.24, not greater than about 0.23, not greater than about0.22, not greater than about 0.21, not greater than about 0.20, notgreater than about 0.19, not greater than about 0.18, not greater thanabout 0.17, not greater than about 0.16, not greater than about 0.15,not greater than about 0.14, not greater than about 0.13, not greaterthan about 0.12, not greater than about 0.11, not greater than about0.10, not greater than about 0.09, not greater than about 0.08, notgreater than about 0.07, not greater than about 0.06, not greater thanabout 0.05, not greater than about 0.04.

Item 12. The shaped abrasive particle of any one of items 2, 3, 4, 5, 6,7, and 8, wherein the body comprises a sharpness-shape-strength factor(3SF) within a range between about 0.7 and about 1.7.

Item 13. The shaped abrasive particle of any one of items 1 and 12,wherein the body has a 3SF of at least about 0.72, at least about 0.75,at least about 0.78, at least about 0.8, at least about 0.82, at leastabout 0.85, at least about 0.88, at least about 0.90, at least about0.92, at least about 0.95, at least about 0.98.

Item 14. The shaped abrasive particle of any one of items 1 and 12, thebody has a 3SF of not greater than about 1.68, not greater than about1.65, not greater than about 1.62, not greater than about 1.6, notgreater than about 1.58, not greater than about 1.55, not greater thanabout 1.52, not greater than about 1.5, not greater than about 1.48, notgreater than about 1.45, not greater than about 1.42, not greater thanabout 1.4, not greater than about 1.38, not greater than about 1.35, notgreater than about 1.32, not greater than about 1.3, not greater thanabout 1.28, not greater than about 1.25, not greater than about 1.22,not greater than about 1.2, not greater than about 1.18, not greaterthan about 1.15, not greater than about 1.12, not greater than about1.1.

Item 15. The shaped abrasive particle of any one of items 1, 4, 5, 6, 7,and 8, wherein the body comprises a strength within a range between atleast about 350 MPa and not greater than about 1500 MPa.

Item 16. The shaped abrasive particle of any one of items 3, 4, and 15,wherein the body comprises a strength of not greater than about 1490MPa, not greater than about 1480 MPa, not greater than about 1470 MPa,not greater than about 1460 MPa, not greater than about 1450 MPa, notgreater than about 1440 MPa, not greater than about 1430 MPa, notgreater than about 1420 MPa, not greater than about 1410 MPa, notgreater than about 1400 MPa, not greater than about 1390 MPa, notgreater than about 1380 MPa, not greater than about 1370 MPa, notgreater than about 1360 MPa, not greater than about 1350 MPa, notgreater than about 1340 MPa, not greater than about 1330 MPa, notgreater than about 1320 MPa, not greater than about 1310 MPa, notgreater than about 1300 MPa, not greater than about 1290 MPa, notgreater than about 1280 MPa, not greater than about 1270 MPa, notgreater than about 1260 MPa, not greater than about 1250 MPa, notgreater than about 1240 MPa, not greater than about 1230 MPa, notgreater than about 1220 MPa, not greater than about 1210 MPa, notgreater than about 1200 MPa, not greater than about 1190 MPa, notgreater than about 1180 MPa, not greater than about 1170 MPa, notgreater than about 1160 MPa, not greater than about 1150 MPa, notgreater than about 1140 MPa, not greater than about 1130 MPa, notgreater than about 1120 MPa, not greater than about 1110 MPa, notgreater than about 1100 MPa, not greater than about 1090 MPa, notgreater than about 1080 MPa, not greater than about 1070 MPa, notgreater than about 1060 MPa, not greater than about 1050 MPa, notgreater than about 1040 MPa, not greater than about 1030 MPa, notgreater than about 1020 MPa, not greater than about 1010 MPa, notgreater than about 1000 MPa, not greater than about 990 MPa, not greaterthan about 980 MPa, not greater than about 970 MPa, not greater thanabout 960 MPa, not greater than about 950 MPa, not greater than about940 MPa, not greater than about 930 MPa, not greater than about 920 MPa,not greater than about 910 MPa, not greater than about 900 MPa, notgreater than about 890 MPa, not greater than about 880 MPa, not greaterthan about 870 MPa, not greater than about 860 MPa, not greater thanabout 850 MPa, not greater than about 840 MPa, not greater than about830 MPa, not greater than about 820 MPa, not greater than about 810 MPa,not greater than about 800 MPa, not greater than about 790 MPa, notgreater than about 780 MPa, not greater than about 770 MPa, not greaterthan about 760 MPa, not greater than about 750 MPa, not greater thanabout 740 MPa, not greater than about 730 MPa, not greater than about720 MPa, not greater than about 710 MPa, not greater than about 700 MPa,not greater than about 690 MPa, not greater than about 680 MPa, notgreater than about 670 MPa, not greater than about 660 MPa, not greaterthan about 650 MPa, not greater than about 640 MPa, not greater thanabout 630 MPa, not greater than about 620 MPa, not greater than about610 MPa, not greater than about 600 MPa, not greater than about 590 MPa,not greater than about 580 MPa, not greater than about 570 MPa, notgreater than about 560 MPa, not greater than about 550 MPa, not greaterthan about 540 MPa, not greater than about 530 MPa, not greater thanabout 520 MPa, not greater than about 510 MPa, not greater than about500 MPa, not greater than about 490 MPa, not greater than about 480 MPa,not greater than about 470 MPa, not greater than about 460 MPa, notgreater than about 450 MPa, not greater than about 440 MPa, not greaterthan about 430 MPa, not greater than about 420 MPa, not greater thanabout 410 MPa, not greater than about 400 MPa.

Item 17. The shaped abrasive particle of any one of items 3, 4, and 15,wherein the body comprises a strength of at least about 360 MPa, atleast about 370 MPa, at least about 380 MPa, at least about 390 MPa, atleast about 400 MPa, at least about 410 MPa, at least about 420 MPa, atleast about 430 MPa, at least about 440 MPa, at least about 450 MPa, atleast about 460 MPa, at least about 470 MPa, at least about 480 MPa, atleast about 490 MPa, at least about 500 MPa, at least about 510 MPa, atleast about 520 MPa, at least about 530 MPa, at least about 540 MPa, atleast about 550 MPa, at least about 560 MPa, at least about 570 MPa, atleast about 580 MPa, at least about 590 MPa, at least about 600 MPa, atleast about 610 MPa, at least about 620 MPa, at least about 630 MPa, atleast about 640 MPa, at least about 650 MPa, at least about 660 MPa, atleast about 670 MPa, at least about 680 MPa, at least about 690 MPa, atleast about 700 MPa, at least about 710 MPa, at least about 720 MPa, atleast about 730 MPa, at least about 740 MPa, at least about 750 MPa, atleast about 760 MPa, at least about 770 MPa, at least about 780 MPa, atleast about 790 MPa, at least about 800 MPa, at least about 810 MPa, atleast about 820 MPa, at least about 830 MPa, at least about 840 MPa, atleast about 850 MPa, at least about 860 MPa, at least about 870 MPa, atleast about 880 MPa, at least about 890 MPa, at least about 900 MPa, atleast about 910 MPa, at least about 920 MPa, at least about 930 MPa, atleast about 940 MPa, at least about 950 MPa, at least about 960 MPa, atleast about 970 MPa, at least about 980 MPa, at least about 990 MPa, atleast about 1000 MPa, at least about 1010 MPa, at least about 1020 MPa,at least about 1030 MPa, at least about 1040 MPa, at least about 1050MPa, at least about 1060 MPa, at least about 1070 MPa, at least about1080 MPa, at least about 1090 MPa, at least about 1100 MPa, at leastabout 1110 MPa, at least about 1120 MPa, at least about 1130 MPa, atleast about 1140 MPa, at least about 1150 MPa, at least about 1160 MPa,at least about 1170 MPa, at least about 1180 MPa, at least about 1190MPa, at least about 1200 MPa, at least about 1210 MPa, at least about1220 MPa, at least about 1230 MPa, at least about 1240 MPa, at leastabout 1250 MPa, at least about 1260 MPa, at least about 1270 MPa, atleast about 1280 MPa, at least about 1290 MPa, at least about 1300 MPa.

Item 18. The shaped abrasive particle of any one of items 1, 2, 4, 5, 6,7, and 8, wherein the body comprises a tip sharpness within a rangebetween not greater than about 80 microns and at least about 1 micron.

Item 19. The shaped abrasive particle of any one of items 3 and 18,wherein the body comprises a tip sharpness of not greater than about 78microns, not greater than about 76 microns, not greater than about 74microns, not greater than about 72 microns, not greater than about 70microns, not greater than about 68 microns, not greater than about 66microns, not greater than about 64 microns, not greater than about 62microns, not greater than about 60 microns, not greater than about 58microns, not greater than about 56 microns, not greater than about 54microns, not greater than about 52 microns, not greater than about 50microns, not greater than about 48 microns, not greater than about 46microns, not greater than about 44 microns, not greater than about 42microns, not greater than about 40 microns, not greater than about 38microns, not greater than about 36 microns, not greater than about 34microns, not greater than about 32 microns, not greater than about 30microns, not greater than about 38 microns, not greater than about 36microns, not greater than about 34 microns, not greater than about 32microns, not greater than about 30 microns, not greater than about 28microns, not greater than about 26 microns, not greater than about 24microns, not greater than about 22 microns, not greater than about 20microns, not greater than about 18 microns, not greater than about 16microns, not greater than about 14 microns, not greater than about 12microns, not greater than about 10 microns.

Item 20. The shaped abrasive particle of any one of items 3 and 18,wherein the body comprises a tip sharpness of at least about 2 microns,at least about 4 microns, at least about 6 microns, at least about 8microns, at least about 10 microns, at least about 12 microns, at leastabout 14 microns, at least about 16 microns, at least about 18 microns,at least about 20 microns, at least about 22 microns, at least about 24microns, at least about 26 microns, at least about 28 microns, at leastabout 30 microns, at least about 32 microns, at least about 34 microns,at least about 36 microns, at least about 38 microns, at least about 40microns, at least about 42 microns, at least about 44 microns, at leastabout 46 microns, at least about 48 microns, at least about 50 microns,at least about 52 microns, at least about 54 microns, at least about 56microns, at least about 58 microns, at least about 60 microns, at leastabout 62 microns, at least about 64 microns, at least about 66 microns,at least about 68 microns, at least about 70 microns.

Item 21. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises an additive, wherein theadditive comprises an oxide, wherein the additive comprises a metalelement, wherein the additive comprises a rare-earth element.

Item 22. The shaped abrasive article of item 21, wherein the additivecomprises a dopant material, wherein the dopant material includes anelement selected from the group consisting of an alkali element, analkaline earth element, a rare earth element, a transition metalelement, and a combination thereof, wherein the dopant materialcomprises an element selected from the group consisting of hafnium,zirconium, niobium, tantalum, molybdenum, vanadium, lithium, sodium,potassium, magnesium, calcium, strontium, barium, scandium, yttrium,lanthanum, cesium, praseodymium, chromium, cobalt, iron, germanium,manganese, nickel, titanium, zinc, and a combination thereof.

Item 23. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises at least about 95 wt % aluminafor the total weight of the body, at least about 95.1 wt %, at leastabout 95.2 wt %, at least about 95.3 wt %, at least about 95.4 wt %, atleast about 95.5 wt %, at least about 95.6 wt %, at least about 95.7 wt%, at least about 95.8 wt %, at least about 95.9 wt %, at least about 96wt %, at least about 96.1 wt %, at least about 96.2 wt %, at least about96.3 wt %, at least about 96.4 wt %, at least about 96.5 wt %, at leastabout 96.6 wt %, at least about 96.7 wt %, at least about 96.8 wt %, atleast about 96.9 wt %, at least about 97 wt %, at least about 97.1 wt %,at least about 97.2 wt %, at least about 975.3 wt %, at least about 97.4wt %, at least about 97.5 wt %.

Item 24. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein body comprises not greater than about 99.5 wt %alumina for the total weight of the body, not greater than about 99.4 wt%, not greater than about 99.3 wt %, not greater than about 99.2 wt %,not greater than about 99.1 wt %, not greater than about 99 wt %, notgreater than about 98.9 wt %, not greater than about 98.8 wt %, notgreater than about 98.7 wt %, not greater than about 98.6 wt %, notgreater than about 98.5 wt %, not greater than about 98.4 wt %, notgreater than about 98.3 wt %, not greater than about 98.2 wt %, notgreater than about 98.1 wt %, not greater than about 98 wt %, notgreater than about 97.9 wt %, not greater than about 97.8 wt %, notgreater than about 97.7 wt %, not greater than about 97.6 wt %, notgreater than about 97.5 wt %.

Item 25. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body consists essentially of alumina.

Item 26. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a polycrystalline materialincluding crystalline grains, wherein the average grain size is notgreater than about 1 micron, not greater than about 0.9 microns, notgreater than about 0.8 microns, not greater than about 0.7 microns, notgreater than about 0.6 microns.

Item 27. The shaped abrasive particle of item 26, wherein the averagegrain size is at least about 0.01 microns, at least about 0.05 microns,at least about 0.06 microns, at least about 0.07 microns, at least about0.08 microns, at least about 0.09 microns, at least about 0.1 microns,at least about 0.12 microns, at least about 0.15 microns, at least about0.17 microns, at least about 0.2 microns.

Item 28. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body is essentially free of a binder, whereinthe body is essentially free of an organic material, wherein the body isessentially free of rare earth elements, wherein the body is essentiallyfree of iron.

Item 29. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body is formed from a seeded sol gel.

Item 30. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a primary aspect ratio ofwidth:length of at least about 1:1 and not greater than about 10:1.

Item 31. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a secondary aspect ratio definedby a ratio of width:height within a range between about 5:1 and about1:1.

Item 32. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a tertiary aspect ratio definedby a ratio of length:height within a range between about 6:1 and about1:1.

Item 33. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a polycrystalline material,wherein the polycrystalline material comprises grains, wherein thegrains are selected from the group of materials consisting of nitrides,oxides, carbides, borides, oxynitrides, diamond, and a combinationthereof, wherein the grains comprise an oxide selected from the group ofoxides consisting of aluminum oxide, zirconium oxide, titanium oxide,yttrium oxide, chromium oxide, strontium oxide, silicon oxide, and acombination thereof, wherein the grains comprise alumina, wherein thegrains consist essentially of alumina.

Item 34. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a two-dimensional polygonalshape as viewed in a plane defined by a length and width, wherein thebody comprises a two dimensional shape having at least 3 exteriorpoints, at least 4 exterior points, at least 5 exterior points, at least6 exterior points, at least 7 exterior points, at least 8 exteriorpoints, at least 9 exterior points.

Item 35. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body is coupled to a substrate as part of afixed abrasive, wherein the fixed abrasive article is selected from thegroup consisting of a bonded abrasive article, a coated abrasivearticle, and a combination thereof.

Item 36. The shaped abrasive particle of item 35, wherein the substrateis a backing, wherein the backing comprises a woven material, whereinthe backing comprises a non-woven material, wherein the backingcomprises an organic material, wherein the backing comprises a polymer,wherein the backing comprises a material selected from the groupconsisting of cloth, paper, film, fabric, fleeced fabric, vulcanizedfiber, woven material, non-woven material, webbing, polymer, resin,phenolic resin, phenolic-latex resin, epoxy resin, polyester resin, ureaformaldehyde resin, polyester, polyurethane, polypropylene, polyimides,and a combination thereof.

Item 37. The shaped abrasive particle of item 36, wherein the backingcomprises an additive selected from the group consisting of catalysts,coupling agents, curants, anti-static agents, suspending agents,anti-loading agents, lubricants, wetting agents, dyes, fillers,viscosity modifiers, dispersants, defoamers, and grinding agents.

Item 38. The shaped abrasive particle of item 36, further comprising anadhesive layer overlying the backing, wherein the adhesive layercomprises a make coat, wherein the make coat overlies the backing,wherein the make coat is bonded directly to a portion of the backing,wherein the make coat comprises an organic material, wherein the makecoat comprises a polymeric material, wherein the make coat comprises amaterial selected from the group consisting of polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, poly vinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 39. The shaped abrasive particle of item 38, wherein the adhesivelayer comprises a size coat, wherein the size coat overlies a portion ofthe plurality of shaped abrasive particles, wherein the size coatoverlies a make coat, wherein the size coat is bonded directly to aportion of the plurality of shaped abrasive particles, wherein the sizecoat comprises an organic material, wherein the size coat comprises apolymeric material, wherein the size coat comprises a material selectedfrom the group consisting of polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, polyvinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 40. The shaped abrasive particle of item 36, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in an open coat, whereinthe open coat comprises a coating density of not greater than about 70particles/cm², not greater than about 65 particles/cm², not greater thanabout 60 particles/cm², not greater than about 55 particles/cm², notgreater than about 50 particles/cm², at least about 5 particles/cm², atleast about 10 particles/cm².

Item 41. The shaped abrasive particle of item 36, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in a closed coat, whereinhaving a closed coat of the blend of shaped abrasive particles on abacking, wherein the closed coat comprises a coating density of at leastabout 75 particles/cm², at least about 80 particles/cm², at least about85 particles/cm², at least about 90 particles/cm², at least about 100particles/cm².

Item 42. The shaped abrasive particle of item 36, wherein the shapedabrasive particle is part of a blend including a plurality of a firsttype of shaped abrasive particles and a third type of abrasive particle,wherein the third type of abrasive particle comprises a shaped abrasiveparticle, wherein the third type of abrasive particle comprises adiluent type of abrasive particle, wherein the diluent type of abrasiveparticle comprises an irregular shape.

Item 43. The shaped abrasive particle of item 42, wherein the blend ofabrasive particles comprises a plurality of shaped abrasive particles,and wherein each shaped abrasive particle of the plurality of shapedabrasive particles is arranged in a controlled orientation relative to abacking, the controlled orientation including at least one of apredetermined rotational orientation, a predetermined lateralorientation, and a predetermined longitudinal orientation.

Item 44. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a length (l), a width (w), and aheight (hi), wherein the length>width, the length>height, and thewidth>height.

Item 45. The shaped abrasive particle of item 44, wherein the height (h)is at least about 70 microns, at least about 80 microns, such as atleast about 90 microns, at least about 100 microns, at least about 100microns, at least about 120 microns, at least about 150 microns, atleast about 175 microns, at least about 200 microns, at least about 225microns, at least about 250 microns, at least about 275 microns at leastabout 300 microns, and not greater than about 3 mm, not greater thanabout 2 mm, not greater than about 1.5 mm, not greater than about 1 mm,not greater than about 800 microns, not greater than about 600 microns,not greater than about 500 microns, not greater than about 475 microns,not greater than about 450 microns, not greater than about 425 microns,not greater than about 400 microns, not greater than about 375 microns,not greater than about 350 microns, not greater than about 325 microns,not greater than about 300 microns, not greater than about 275 microns,not greater than about 250 microns.

Item 46. The shaped abrasive particle of item 44, wherein the width isat least about 200 microns, at least about 250 microns, at least about300 microns, at least about 350 microns, at least about 400 microns, atleast about 450 microns, at least about 500 microns, at least about 550microns, at least about 600 microns, at least about 700 microns, atleast about 800 microns, at least about 900 microns, and not greaterthan about 4 mm, not greater than about 3 mm, not greater than about 2.5mm, not greater than about 2 mm.

Item 47. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a percent flashing of notgreater than about 40%, not greater than about 35%, not greater thanabout 30%, not greater than about 25%, not greater than about 20%, notgreater than about 18%, not greater than about 15%, not greater thanabout 12%, not greater than about 10%, not greater than about 8%, notgreater than about 6%, not greater than about 4%.

Item 48. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises a dishing value (d) of notgreater than about 2, not greater than about 1.9, not greater than about1.8, not greater than about 1.7, not greater than about 1.6, not greaterthan about 1.5, not greater than about 1.2, and at least about 0.9, atleast about 1.0.

Item 49. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the shaped abrasive particle is part of a pluralityof a first type of shaped abrasive particles, wherein a majority of thefirst type of shaped abrasive particles are coupled to the backing in aside orientation, wherein at least about 55% of the shaped abrasiveparticles of the plurality of shaped abrasive particles are coupled tothe backing in a side orientation, at least about 60%, at least about65%, at least about 70%, at least about 75%, at least about 77%, atleast about 80%, and not greater than about 99%, not greater than about95%, not greater than about 90%, not greater than about 85%.

Item 50. The shaped abrasive particle of any one of items 1, 2, 3, 5, 6,7, and 8, wherein at least one side surface of the body has apartially-concave shape.

Item 51. The shaped abrasive particle of any one of items 1, 2, 3, 4, 7,and 8, wherein a first portion of the side surface extends between afirst corner and a second corner of the body that are adjacent to eachother, and wherein the first portion of the side surface comprises afirst curved section joined to a first linear section.

Item 52. The shaped abrasive particle of any one of items 4, 5, and 51,wherein the first linear section comprises a first linear section length(Ll1) and the first curved section comprises a first curved sectionlength (Lc1).

Item 53. The shaped abrasive particle of item 52, wherein Lc1≥Ll1.

Item 54. The shaped abrasive particle of item 52, wherein Ll1≥Lc1.

Item 55. The shaped abrasive particle of item 52, further comprising alength factor (Ll1/Lc1) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 56. The shaped abrasive particle of item 55, wherein the lengthfactor (Ll1/Lc1) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2.

Item 57. The shaped abrasive particle of item 52, further comprising alength factor (Lc1/Ll1) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 58. The shaped abrasive particle of item 57, wherein the lengthfactor (Lc1/Ll1) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2.

Item 59. The shaped abrasive particle of any one of items 4, 5, and 51,wherein the first portion of the side surface further comprises a secondlinear section joined to the first curved section.

Item 60. The shaped abrasive particle of item 59, wherein the secondlinear section comprises a length (Ll2), and wherein Ll1 issubstantially equal to Ll2.

Item 61. The shaped abrasive particle of item 59, wherein Lc1≥Ll2.

Item 62. The shaped abrasive particle of item 59, further comprising alength factor (Ll2/Lc1) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 63. The shaped abrasive particle of item 62, wherein the lengthfactor (Ll2/Lc1) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2.

Item 64. The shaped abrasive particle of item 59, further comprising alength factor (Lc1/Ll2) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 65. The shaped abrasive particle of item 59, wherein the lengthfactor (Lc1/Ll2) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2.

Item 66. The shaped abrasive particle of item 59, further comprising alinear sum factor ((Ll1+Ll2)/Lc1) of not greater than about 1, notgreater than about 0.95, not greater than about 0.9, not greater thanabout 0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 67. The shaped abrasive particle of item 66, wherein the linear sumfactor ((Ll1+Ll2)/Lc1) is at least about 0.05, at least about 0.1, atleast about 0.15, at least about 0.2.

Item 68. The shaped abrasive particle of item 59, further comprising aninverse linear sum factor (Lc1/(Ll1+Ll2)) of not greater than about 1,not greater than about 0.95, not greater than about 0.9, not greaterthan about 0.85, not greater than about 0.8, not greater than about0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, not greater than about 0.05.

Item 69. The shaped abrasive particle of item 68, wherein the inverselinear sum factor (Lc1/(Ll1+Ll2)) is at least about 0.05, at least about0.1, at least about 0.15, at least about 0.2.

Item 70. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,7, and 8, wherein a first portion of the side surface comprises a firstcurved section joined to a first linear section and defining an interiorcorner defining an obtuse angle.

Item 71. The shaped abrasive particle of any one of items 6 and 70,wherein the obtuse angle has a value between at least about 92 degreesand not greater than about 178 degrees, wherein the obtuse angle is atleast about 94 degrees, at least about 96 degrees, at least about 98degrees, at least about 100 degrees, at least about 102 degrees, atleast about 104 degrees, at least about 106 degrees, at least about 108degrees, at least about 110 degrees, at least about 112 degrees, atleast about 124 degrees, at least about 126 degrees, at least about 128degrees, at least about 120 degrees, at least about 122 degrees, atleast about 124 degrees, at least about 126 degrees, at least about 128degrees, at least about 130 degrees, at least about 132 degrees, atleast about 134 degrees, at least about 136 degrees, at least about 138degrees, at least about 140 degrees.

Item 72. The shaped abrasive particle of item 71, wherein the obtuseangle is not greater than about 176 degrees, not greater than about 174degrees, not greater than about 172 degrees, not greater than about 170degrees, not greater than about 168 degrees, not greater than about 166degrees, not greater than about 164 degrees, not greater than about 162degrees, not greater than about 160 degrees, not greater than about 158degrees, not greater than about 156 degrees, not greater than about 154degrees, not greater than about 152 degrees, not greater than about 150degrees, not greater than about 148 degrees, not greater than about 146degrees, not greater than about 144 degrees, not greater than about 142degrees, not greater than about 140 degrees.

Item 73. The shaped abrasive particle of any one of items 6 and 70,wherein the first portion of the side surface comprises a first interiorcorner and a second interior corner, wherein the first interior cornerand the second interior corner are spaced apart from each other atopposite ends of the first curved section, wherein the first interiorcorner is disposed at the edge between the first linear section and thefirst curved section, and wherein the second interior corner is disposedat an edge between the first curved section and a second linear section.

Item 74. The shaped abrasive particle of any one of items 1, 2, 3, 5, 6,7, and 8, wherein the body comprises a first portion of the side surfacehaving a partially-concave shape.

Item 75. The shaped abrasive particle of any one of items 4 and 74,wherein the partially-concave shape comprises a first curved sectionhaving a first curved section length (Lc1) that is a fraction of a totallength of the first portion (Lfp1) of the side surface.

Item 76. The shaped abrasive particle of item 75, further comprising alength factor (Lc1/Lfp) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 77. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section has at least two distinct curvatures.

Item 78. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section defines an arc having a single distinctcurvature.

Item 79. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section defines a concave shape.

Item 80. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section is disposed between a first linearsection and a second linear section, wherein the first linear sectionterminates at a first end at a first external corner of the body,extends along the first portion of the side surface for a length (Ll1),and terminates at a second end at the first curved section and defines afirst internal corner, wherein the second linear section terminates atfirst end at a second external corner of the body, extends along thefirst portion of the side surface for a length (Ll2), and terminates ata second end at the first curved section and defines a second internalcorner.

Item 81. The shaped abrasive particle of any one of items 4 and 74,wherein the length of the first linear section (Ll1) is a fraction of atotal length of the first portion of the side surface (Lfp), furthercomprising a length factor (Ll1/Lfp) of not greater than about 1, notgreater than about 0.95, not greater than about 0.9, not greater thanabout 0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 82. The shaped abrasive particle of any one of items 4 and 74,wherein the length of the second linear section (Ll2) is a fraction of atotal length of the first portion of the side surface (Lfp), furthercomprising a length factor (Ll2/Lfp) of not greater than about 1, notgreater than about 0.95, not greater than about 0.9, not greater thanabout 0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 83. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section has a radius of curvature (Rc1) and thebody comprises a width (w), and wherein the radius of curvature (Rc1) isat least half the width of the body, at least about 0.8 times the widthof the body, at least 1.5 times the width of the body, at least 2 timesthe width of the body, and wherein the radius of curvature (Rc1) is notgreater than about 50 times the width.

Item 84. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section has a first end spaced apart from afirst external corner of the body.

Item 85. The shaped abrasive particle of any one of items 4 and 74,wherein the first curved section has a second end spaced apart from asecond external corner of the body.

Item 86. The shaped abrasive particle of any one of items 4 and 74,wherein the body further comprises a second curved section

Item 87. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises:

a second portion of the side surface extending between a second cornerand a third corner of the body that are adjacent to each other, andwherein the second portion of the side surface comprises a second curvedsection joined to a third linear section.

Item 88. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein

a third portion of the side surface extending between a first corner anda third corner of the body that are adjacent to each other, and whereinthe third portion of the side surface comprises a third curved sectionjoined to a fifth linear section.

Item 89. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, 7, and 8, wherein the body comprises at least a first portion of theside surface having at least a first curved section, a second portion ofthe side surface separated from the first portion of the side surface byat least one external corner, the second portion comprising at least asecond curved section, and a third portion of the side surface separatedfrom the first portion of the side surface by at least one externalcorner and further separated from the second portion of the side surfaceby at least one external corner, wherein the third portion comprises atleast a third curved section.

Item 90. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, and 8, wherein the body is a hybrid polygonal shape having a sum ofthe external corners of substantially 180 degrees, and furthercomprising a first portion of the side surface having a first curvedsection.

Item 91. The shaped abrasive particle of any one of items 1, 2, 3, 4, 5,6, and 7, wherein the body comprises a first maximum tip width (Wt1)that defines a maximum width of a first arm of the body, and wherein thefirst maximum tip width (Wt1) is disposed a distance from a firstterminal end of the first arm and between a midpoint of the body and thefirst terminal end.

Item 92. The shaped abrasive particle of any one of items 8 and 91,wherein the first maximum tip width (Wt1) defines a maximum tip widthlocation along an axis of the first arm, and wherein the first armcomprises a first tip length (Ltip1) extending between the terminal endof the arm and the maximum tip width location, wherein the first tiplength (Ltip1) is a fraction of a total length of the first arm (Larm1)defined as a distance between the terminal end of the arm and themidpoint of the body, wherein the first tip length (Ltip1) is at leastabout 0.01 (Larm1), at least about 0.02 (Larm1), at least about 0.03(Larm1), at least about 0.04 (Larm1), at least about 0.05 (Larm1), atleast about 0.06 (Larm1), at least about 0.07 (Larm1), at least about0.08 (Larm1), at least about 0.09 (Larm1), at least about 0.1 (Larm1),at least about 0.12 (Larm1), at least about 0.15 (Larm1), at least about0.18 (Larm1), at least about 0.2 (Larm1), at least about 0.22 (Larm1),at least about 0.25 (Larm1), at least about 0.28 (Larm1), at least about0.3 (Larm1), at least about 0.32 (Larm1), at least about 0.35 (Larm1),at least about 0.38 (Larm1), at least about 0.4 (Larm1).

Item 93. The shaped abrasive particle of item 92, wherein the first tiplength (Ltip1) is not greater than about 0.95 (Larm1), not greater thanabout 0.9 (Larm1), not greater than about 0.85 (Larm1), not greater thanabout 0.8 (Larm1), not greater than about 0.75 (Larm1), not greater thanabout 0.7 (Larm1), not greater than about 0.65 (Larm1), not greater thanabout 0.6 (Larm1), not greater than about 0.55 (Larm1), not greater thanabout 0.5 (Larm1), not greater than about 0.45 (Larm1).

Item 94. The shaped abrasive particle of any one of items 8 and 91,wherein the body further comprises a first throat width (Wth1) definingthe narrowest portion of the first arm between a first maximum tip widthlocation of the first arm and the midpoint, wherein the first throatwidth (Wth1) defines a first throat width location along an axis of thefirst arm, wherein the first throat location is closer to the midpointthan the first maximum tip width location.

Item 95. The shaped abrasive particle of item 94, wherein the firstthroat location defines a first throat length (Lth1) extending betweenthe first throat location and the first maximum tip width location,wherein the first throat length (Lth1) is a fraction of a total lengthof the first arm (Larm1), wherein the first throat length (Lth1) is atleast about 0.01 (Larm1), at least about 0.02 (Larm1), at least about0.03 (Larm1), at least about 0.04 (Larm1), at least about 0.05 (Larm1),at least about 0.06 (Larm1), at least about 0.07 (Larm1), at least about0.08 (Larm1), at least about 0.09 (Larm1), at least about 0.1 (Larm1),at least about 0.12 (Larm1), at least about 0.15 (Larm1), at least about0.18 (Larm1), at least about 0.2 (Larm1), at least about 0.22 (Larm1),at least about 0.25 (Larm1), at least about 0.28 (Larm1), at least about0.3 (Larm1), at least about 0.32 (Larm1), at least about 0.35 (Larm1),at least about 0.38 (Larm1), at least about 0.4 (Larm1).

Item 96. The shaped abrasive particle of item 95, wherein the firstthroat length (Lth1) is not greater than about 0.95 (Larm1), not greaterthan about 0.9 (Larm1), not greater than about 0.85 (Larm1), not greaterthan about 0.8 (Larm1), not greater than about 0.75 (Larm1), not greaterthan about 0.7 (Larm1), not greater than about 0.65 (Larm1), not greaterthan about 0.6 (Larm1), not greater than about 0.55 (Larm1), not greaterthan about 0.5 (Larm1), not greater than about 0.45 (Larm1).

Item 97. The shaped abrasive particle of item 94, wherein the firstthroat width (Wth1) is less than the first maximum tip width (Wt1).

Item 98. The shaped abrasive particle of item 97, wherein the firstthroat width (Wth1) is not greater than about 0.95 (Wt1), not greaterthan about 0.9 (Wt1), not greater than about 0.85 (Wt1), not greaterthan about 0.8 (Wt1), not greater than about 0.75 (Wt1), not greaterthan about 0.7 (Wt1), not greater than about 0.65 (Wt1), not greaterthan about 0.6 (Wt1), not greater than about 0.55 (Wt1), not greaterthan about 0.5 (Wt1), not greater than about 0.45 (Wt1).

Item 99. The shaped abrasive particle of item 98, wherein the firstthroat width (Wth1) is at least about 0.01 (Wt1), at least about 0.05(Wt1), at least about 0.08 (Wt1), at least about 0.1 (Wt1), at leastabout 0.12 (Wt1), at least about 0.15 (Wt1), at least about 0.18 (Wt1),at least about 0.2 (Wt1), at least about 0.22 (Wt1), at least about 0.25(Wt1), at least about 0.28 (Wt1), at least about 0.3 (Wt1), at leastabout 0.32 (Wt1), at least about 0.35 (Wt1), at least about 0.38 (Wt1),at least about 0.4 (Wt1), at least about 0.42 (Wt1), at least about 0.45(Wt1), at least about 0.48 (Wt1), at least about 0.5 (Wt1).

Item 100. The shaped abrasive particle of any one of items 8 and 91,further comprising:

a second maximum tip width (Wt2) that defines a maximum width of asecond arm of the body, and wherein the second maximum tip width (Wt2)is disposed between a second terminal end of the second arm and themidpoint of the body; and

a third maximum tip width (Wt3) that defines a maximum width of a thirdarm of the body, and wherein the third maximum tip width (Wt3) isdisposed between a third terminal end of the third arm and the midpointof the body.

Item 101. The shaped abrasive particle of item 100, further comprising:

wherein the body further comprises a second throat width (Wth2) definingthe narrowest portion of the second arm between a second maximum tipwidth location of the second arm and the midpoint, wherein the secondthroat width (Wth2) defines a second throat width location along an axisof the second arm and wherein the second throat location is closer tothe midpoint than the second maximum tip width location; and

wherein the body further comprises a third throat width (Wth3) definingthe narrowest portion of the third arm between the third maximum widthlocation of the third arm and the midpoint, wherein the third throatwidth (Wth3) defines a third throat width location along an axis of thethird arm and wherein the third throat location is closer to themidpoint than the third maximum tip width location.

Item 102. A method of making a shaped abrasive particle comprisingforming a body of a shaped abrasive comprising at least one of:

i) selecting a material having a predetermined strength within a rangebetween at least about 350 MPa and not greater than about 1500 MPa andforming the body of the shaped abrasive particle with a predeterminedtip sharpness and predetermined Shape Index based upon the predeterminedstrength;

ii) selecting a predetermined Shape Index of the body of the shapedabrasive particle within a range between at least about 0.01 and notgreater than about 0.49 and forming the body with the predetermined tipsharpness and the predetermined strength based upon the predeterminedShape Index; and

iii) selecting a predetermined tip sharpness of a body of the shapedabrasive particle within a range between at least about 1 micron and notgreater than about 80 microns, and forming the body of a shaped abrasiveparticle with a predetermined Shape Index and a predetermined strengthbased upon the predetermined tip sharpness.

Item 103. The method of item 102, wherein forming comprises controllingthe interrelationship of grain features selected from the group ofpredetermined tip sharpness, predetermined Shape Index, andpredetermined strength of the body of the shaped abrasive particle toinfluence a self-sharpening behavior of the shaped abrasive particle.

Item 104. The method of item 102, wherein controlling theinterrelationship of grain features comprises forming the body with asharpness-shape-strength factor (3SF) within a range between about 0.7and about 1.7.

Item 105. The method of item 102, wherein forming comprises a methodselected from the group consisting of depositing, printing, extruding,molding, casting, pressing, punching, sectioning, and a combinationthereof.

Item 106. The method of item 102, wherein the body comprises a firstmajor surface, a second major surface, and a side surface extendingbetween the first major surface and the second major surface, andwherein the body comprises at least one of:

a first portion of the side surface having a partially-concave shape;

a first portion of the side surface extending between a first corner andsecond corner of the body that are adjacent to each other, and whereinthe first portion of the side surface comprises a first curved sectionjoined to a first linear section;

a first portion of the side surface comprising a first curved sectionjoined to a first linear section and defining an interior cornerdefining an obtuse angle;

a hybrid polygonal shape having a sum of the external corners ofsubstantially 180 degrees, and further comprising a first portion of theside surface having a first curved section;

a first maximum tip width (Wt1) that defines a maximum width of a firstarm of the body, and wherein the first maximum tip width (Wt1) isdisposed between a first terminal end of the first arm and a midpoint ofthe body; and a combination thereof.

Item 107. A fixed abrasive comprising abrasive particles overlying asubstrate, wherein the fixed abrasive comprises a half-life/initialenergy factor of not greater than 1.

Item 108. A fixed abrasive comprising abrasive particles overlying asubstrate, wherein the fixed abrasive comprises a minimum specificgrinding energy factor of at least 5%.

Item 109. The fixed abrasive article of item 108, wherein the fixedabrasive comprises a half-life/initial energy factor of not greater than1.

Item 110. The fixed abrasive article of any one of items 107 and 109,wherein the fixed abrasive comprises a half-life/initial energy factorof not greater than 1.20 or not greater than 1.16 or not greater than1.10 or not greater than 1.06 or not greater than 1.00 or not greaterthan 0.99 or not greater than 0.98 or not greater than 0.97 or notgreater than 0.96 or not greater than 0.95 or not greater than 0.94 ornot greater than 0.93 or not greater than 0.92 or not greater than 0.91or not greater than 0.9 or not greater than 0.89 or not greater than0.88 or not greater than 0.87 or not greater than 0.86 or not greaterthan 0.85 or not greater than 0.84 or not greater than 0.83 or notgreater than 0.82 or not greater than 0.81 or not greater than 0.8 ornot greater than 0.79 or not greater than 0.78 or not greater than 0.77or not greater than 0.76 or not greater than 0.75 or not greater than0.74 or not greater than 0.73 or not greater than 0.72 or not greaterthan 0.71 or not greater than 0.7 or not greater than 0.69 or notgreater than 0.68 or not greater than 0.67 or not greater than 0.66 ornot greater than 0.65 or not greater than 0.64 or not greater than 0.63or not greater than 0.62 or not greater than 0.61 or not greater than0.6 or not greater than 0.55 or not greater than 0.5.

Item 111. The fixed abrasive article of item 110, wherein the fixedabrasive comprises a half-life/initial energy factor of at least 0.01 orat least 0.1 or at least 0.15 or at least 0.2 or at least 0.25 or atleast 0.3 or at least 0.35 or at least 0.4 or at least 0.45 or at least0.5 or at least 0.55 or at least 0.6 or at least 0.65 or at least 0.7 orat least 0.75 or at least 0.8 or at least 0.85 or at least 0.9 or atleast 0.95 or at least 0.99 or at least 1 or at least 1.01.

Item 112. The fixed abrasive article of item 107, wherein the fixedabrasive comprises a minimum specific grinding energy factor of at least5%.

Item 113. The fixed abrasive article of any one of items 108 and 112,wherein the fixed abrasive comprises a minimum specific grinding energyfactor of at least 5.5% or at least 6% or at least 6.5% or at least 7%or at least 7.5% or at least 8% or at least 8.5% or at least 9% or atleast 9.5% or at least 10% or at least 10.5% or at least 11% or at least11.5% or at least 12% or at least 12.5% or at least 13% or at least13.5% or at least 14% or at least 14.5%.

Item 114. The fixed abrasive article of item 113, wherein the fixedabrasive comprises a minimum specific grinding energy factor of notgreater than 60% or not greater than 50% or not greater than 40% or notgreater than 30% or not greater than 25% or not greater than 20% or notgreater than 18% or not greater than 15%.

Item 115. The fixed abrasive article of any one of items 107 and 108,further comprising a coated abrasive article defining a single layer ofthe abrasive particles coupled to a major surface of the substrate.

Item 116. The fixed abrasive article of any one of items 107 and 108,wherein the substrate is a backing, wherein the backing comprises awoven material, wherein the backing comprises a non-woven material,wherein the backing comprises an organic material, wherein the backingcomprises a polymer, wherein the backing comprises a material selectedfrom the group consisting of cloth, paper, film, fabric, fleeced fabric,vulcanized fiber, woven material, non-woven material, webbing, polymer,resin, phenolic resin, phenolic-latex resin, epoxy resin, polyesterresin, urea formaldehyde resin, polyester, polyurethane, polypropylene,polyimides, and a combination thereof.

Item 117. The fixed abrasive article of item 116, wherein the backingcomprises an additive selected from the group consisting of catalysts,coupling agents, curants, anti-static agents, suspending agents,anti-loading agents, lubricants, wetting agents, dyes, fillers,viscosity modifiers, dispersants, defoamers, and grinding agents.

Item 118. The fixed abrasive article of item 116, further comprising anadhesive layer overlying the backing, wherein the adhesive layercomprises a make coat, wherein the make coat overlies the backing,wherein the make coat is bonded directly to a portion of the backing,wherein the make coat comprises an organic material, wherein the makecoat comprises a polymeric material, wherein the make coat comprises amaterial selected from the group consisting of polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, poly vinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 119. The fixed abrasive article of item 118, wherein the adhesivelayer comprises a size coat, wherein the size coat overlies a portion ofthe plurality of shaped abrasive particles, wherein the size coatoverlies a make coat, wherein the size coat is bonded directly to aportion of the plurality of shaped abrasive particles, wherein the sizecoat comprises an organic material, wherein the size coat comprises apolymeric material, wherein the size coat comprises a material selectedfrom the group consisting of polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, polyvinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 120. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include shaped abrasive particles.

Item 121. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a plurality of a first type ofshaped abrasive particles, wherein a majority of the first type ofshaped abrasive particles are coupled to the backing in an open coat,wherein the open coat comprises a coating density of not greater thanabout 70 particles/cm².

Item 122. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a plurality of a first type ofshaped abrasive particles, wherein a majority of the first type ofshaped abrasive particles are coupled to the backing in a closed coat,wherein having a closed coat of the blend of shaped abrasive particleson a backing, wherein the closed coat comprises a coating density of atleast about 75 particles/cm².

Item 123. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a blend including a plurality ofa first type of shaped abrasive particles and a second type of abrasiveparticle, wherein the second type of abrasive particle comprises ashaped abrasive particle, wherein the second type of abrasive particlecomprises a diluent type of abrasive particle, wherein the diluent typeof abrasive particle comprises an irregular shape.

Item 124. The fixed abrasive article of item 123, wherein the blend ofabrasive particles comprises a plurality of shaped abrasive particles,and wherein each shaped abrasive particle of the plurality of shapedabrasive particles is arranged in a controlled orientation relative to abacking, the controlled orientation including at least one of apredetermined rotational orientation, a predetermined lateralorientation, and a predetermined longitudinal orientation.

Item 125. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises an average tip sharpnesswithin a range between not greater than about 80 microns and at leastabout 1 micron, a Shape Index within a range between at least about 0.01and not greater than about 0.47, and the body comprises a strength of atleast about 350 MPa and not greater than about 1500 MPa.

Item 126. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein a first portion of the side surface has apartially-concave shape.

Item 127. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein a first portion of the side surface extendsbetween a first corner and second corner of the body that are adjacentto each other, and wherein the first portion of the side surfacecomprises a first curved section joined to a first linear section.

Item 128. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein a first portion of the side surface comprises afirst curved section joined to a first linear section and defining aninterior corner defining an obtuse angle.

Item 129. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body is a hybrid polygonal shape having a sumof the external corners of substantially 180 degrees, and furthercomprising a first portion of the side surface having a first curvedsection.

Item 130. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises a first maximum tip width(Wt1) that defines a maximum width of a first arm of the body, andwherein the first maximum tip width (Wt1) is disposed a distance from afirst terminal end of the first arm and between a midpoint of the bodyand the first terminal end.

Item 131. The fixed abrasive article of any one of items 107 and 108,wherein at least a majority of the abrasive particles include shapedabrasive particles.

Item 132. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body including a first portion of the side surface having apartially-concave shape.

Item 133. The fixed abrasive article of item 132, wherein thepartially-concave shape comprises a first curved section having a firstcurved section length (Lc1) that is a fraction of a total length of afirst portion (Lfp1) of the side surface.

Item 134. The fixed abrasive article of item 133, further comprising alength factor (Lc1/Lfp) of not greater than about 1 or not greater thanabout 0.95 or not greater than about 0.9 or not greater than about 0.85or not greater than about 0.8 or not greater than about 0.75 or notgreater than about 0.7 or not greater than about 0.65 or not greaterthan about 0.6 or not greater than about 0.55 or not greater than about0.5 or not greater than about 0.45 or not greater than about 0.4 or notgreat not greater than about 0.35 or not greater than about 0.3 or notgreater than about 0.35 or not greater than about 0.3 or not greaterthan about 0.25 or not greater than about 0.2 or not greater than about0.15 or not greater than about 0.1 or not greater than about 0.05.

Item 135. The fixed abrasive article of item 133, wherein the firstcurved section has at least two distinct curvatures.

Item 136. The fixed abrasive article of item 133, wherein the firstcurved section defines an arc having a single distinct curvature.

Item 137. The fixed abrasive article of item 133, wherein the firstcurved section defines a concave shape.

Item 138. The fixed abrasive article of item 133, wherein the firstcurved section is disposed between a first linear section and a secondlinear section, wherein the first linear section terminates at a firstend at a first external corner of the body, extends along the firstportion of the side surface for a length (Ll1), and terminates at asecond end at the first curved section and defines a first internalcorner, wherein the second linear section terminates at first end at asecond external corner of the body, extends along the first portion ofthe side surface for a length (Ll2), and terminates at a second end atthe first curved section and defines a second internal corner.

Item 139. The fixed abrasive article of item 133, wherein the firstcurved section has a radius of curvature (Rc1) and the body comprises awidth (w), and wherein the radius of curvature (Rc1) is at least halfthe width of the body or at least about 0.8 times the width of the bodyor at least 1.5 times the width of the body or at least 2 times thewidth of the body, and wherein the radius of curvature (Rc1) is notgreater than about 50 times the width.

Item 140. The fixed abrasive article of item 133, wherein the firstcurved section has a radius of curvature (Rc1) that is not greater than4 mm or not greater than 3 mm or not greater than 2.5 mm or at least0.01 mm or at least 0.1 mm or at least 0.5 mm or at least 0.8 mm or atleast 1 mm or at least 1.1 mm or at least 1.5 mm.

Item 141. The fixed abrasive article of any one of items 107 and 108,wherein the abrasive particles include a shaped abrasive particle havinga body comprising a first maximum tip width (Wt1) that defines a maximumwidth of a first arm of the body, and wherein the first maximum tipwidth (Wt1) is disposed a distance from a first terminal end of thefirst arm and between a midpoint of the body and the first terminal end.

Item 142. The fixed abrasive article of item 141, wherein the firstmaximum tip width (Wt1) defines a maximum tip width location along anaxis of the first arm, and wherein the first arm comprises a first tiplength (Ltip1) extending between the terminal end of the arm and themaximum tip width location.

Item 143. The fixed abrasive article of item 141, wherein the first tiplength (Ltip1) is a fraction of a total length of the first arm (Larm1)defined as a distance between the terminal end of the arm and themidpoint of the body, wherein the first tip length (Ltip1) is at leastabout 0.01 (Larm1).

Item 144. The fixed abrasive article of item 141, wherein the first tiplength (Ltip1) is not greater than about 0.95 (Larm1).

Item 145. The fixed abrasive article of any one of items 107 and 108,wherein the body further comprises a first throat width (Wth1) definingthe narrowest portion of a first arm between a first maximum tip widthlocation of the first arm and a midpoint, wherein the first throat width(Wth1) defines a first throat width location along an axis of the firstarm, and wherein the first throat location is closer to the midpointthan the first maximum tip width location.

Item 146. The fixed abrasive article of item 145, wherein the firstthroat location defines a first throat length (Lth1) extending betweenthe first throat location and the first maximum tip width location, andwherein the first throat length (Lth1) is a fraction of a total lengthof the first arm (Larm1), and wherein the first throat length (Lth1) isat least about 0.01 (Larm1).

Item 147. The fixed abrasive article of item 145, wherein the firstthroat length (Lth1) is not greater than about 0.95 (Larm1).

Item 148. The fixed abrasive article of item 145, wherein the firstthroat width (Wth1) is less than the first maximum tip width (Wt1),wherein the first throat width (Wth1) is not greater than about 0.95(Wt1).

Item 149. The fixed abrasive article of item 148, wherein the firstthroat width (Wth1) is at least about 0.01 (Wt1).

What is claimed is:
 1. A shaped abrasive particle comprising: a bodycomprising: a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises a two-dimensional shape as viewed ina plane parallel to the first major surface; wherein a first portion ofthe side surface comprises a first section defining an indentation intothe body, the first section extending between a first linear section anda second linear section, wherein the first linear section terminates ata first external corner of the body and the second linear sectionterminates at a second external corner of the body, and wherein thefirst section does not intersect the first external corner or secondexternal corner; and wherein the first section comprises an apex havingcurvature defining a radius of curvature of not greater than 50 times awidth of a body.
 2. The shaped abrasive particle of claim 1, wherein theradius of curvature is at least 0.01 mm.
 3. The shaped abrasive particleof claim 1, wherein the indentation extends into the body toward amidpoint of the body.
 4. The shaped abrasive particle of claim 1,wherein the first section comprises a maximum depth that is a fractionof the total width of the body.
 5. The shaped abrasive particle of claim1, wherein the first section defines a continuous concave curvature. 6.The shaped abrasive particle of claim 1, wherein the first portioncomprises a second section defining an indentation into the body.
 7. Theshaped abrasive particle of claim 6, wherein the second section extendsbetween the first linear section and a second linear section.
 8. Theshaped abrasive particle of claim 6, wherein the second sectioncomprises an apex having curvature defining a radius of curvature of notgreater than 50 times a width of a body.
 9. The shaped abrasive particleof claim 8, wherein the second section comprises a radius of curvatureof at least 0.01 mm.
 10. The shaped abrasive particle of claim 8,wherein the first section comprises a radius of curvature greater thanthe radius of curvature of the second section.
 11. The shaped abrasiveparticle of claim 1, wherein the first section comprises a curvaturelength that is a fraction of a total length of the first portion (Lfp1)of the side surface.
 12. The shaped abrasive particle of claim 1,wherein the curvature extends for a full thickness of the body of theshaped abrasive particle along the side surface in a direction betweenthe first and second major surfaces.
 13. The shaped abrasive particle ofclaim 1, wherein the first section and the first linear section define afirst interior corner defining an obtuse angle.
 14. A plurality ofshaped abrasive particles, each of the shaped abrasive particles havingthe features of claim 1, and wherein the plurality of shaped abrasiveparticles are part of a fixed abrasive article selected from the groupconsisting of a bonded abrasive article, a coated abrasive article, or acombination thereof.
 15. A shaped abrasive particle comprising: a bodycomprising: a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises a two-dimensional shape as viewed ina plane parallel to the first major surface; wherein a first portion ofthe side surface comprises a first section defining an indentation intothe body, the first section extending between a first linear section anda second linear section, wherein the first linear section terminates ata first external corner of the body and the second linear sectionterminates at a second external corner of the body, and wherein thefirst section does not intersect the first external corner or secondexternal corner; and wherein the first section and the first linearsection define a first interior corner defining an obtuse angle.
 16. Theshaped abrasive particle of claim 15, wherein the obtuse angle is atleast about 92 degrees.
 17. The shaped abrasive particle of claim 15,wherein the obtuse angle is at least about 120 degrees.
 18. The shapedabrasive particle of claim 15, wherein the first section comprises amaximum depth that is a fraction of the total width of the body.
 19. Theshaped abrasive particle of claim 15, wherein the first portioncomprises a second section defining an indentation into the body.
 20. Aplurality of shaped abrasive particles, each of the shaped abrasiveparticles having the features of claim 15, and wherein the plurality ofshaped abrasive particles are part of a fixed abrasive article selectedfrom the group consisting of a bonded abrasive article, a coatedabrasive article, or a combination thereof.